You ask how is 1 Corinthians 8:6
yet for us there is one God, the Father, of whom are all things, and we for Him; and one Lord Jesus Christ, through whom are all things, and through whom we live.
to be understood with Trinitarian theology? In particular, you state "Paul is saying that the one being of God is shared only by the Father." Trinitarians have never seen 1 Corinthians 8:6 that way. They have focused on the unity of the Father and the Son, the Logos, in their action in creation. What was shocking to both the Jewish and pagan converts was not that Paul placed the Father and the Son/Logos on equal level but that he identified the Logos with the man Jesus Christ. Trinitarians do not understand 1 Corinthians 8:6 as separating the Father from the Son, but rather believe that Paul is placing the Lord Jesus Christ on equal footing with the one God, the Father by identifying Him with the Logos.
The Nicene Creed
how can we as Trinitarians reconcile the fact that "there is one God, the Father" with the doctrine of the trinity
There is no need to reconcile it with Trinitarian theology since it is the correct expression of Trinitarian theology. In particular the Church Fathers gathered at Nicaea for the First Ecumenical Council felt this was the most appropriate way to refer to the Father and the Son. Notice the how the original 325 version of the Nicene Creed lines up with 1 Corinthians 8:6:
1 Corinthians 8:6 Nicene Creed (325)
There is one God, the Father We believe in one God, the Father Almighty
of whom are all things Maker of all things visible and invisible.
and we for Him
and one Lord Jesus Christ And in one Lord Jesus Christ
the Son of God,
begotten of the Father
that is, of the essence of the Father,
God of God, Light of Light, very God of very God,
begotten, not made,
being of one substance with the Father
through whom are all things, By whom all things were made both in heaven and on earth
and through whom we live.
Who for us men, and for our salvation,
came down and was incarnate and was made man.
He suffered, and the third day he rose again, ascended into heaven
From thence he shall come to judge the quick and the dead.
And in the Holy Ghost.
While the Nicene Fathers apparently felt that words of 1 Corinthians 8:6 accurately stated their position, they did find in necessary to clarify the relationship between the Father and the Son. Why? Because the Arians presented this same verse as a proof text. They asserted, somewhat like your question implies, that only the Father is God and that Christ something else, something lesser, a Lord. But this in no way implies that they held a Unitarian understanding of 1 Corinthians 8:6. Arius and his followers believed that Christ was God as well. Arius himself wrote in a letter to his fellow Arian Eusebius, Bishop of Nicomedia:
But we say and believe, and have taught, and do teach, that the Son is not unbegotten, nor in any way part of the unbegotten; and that He does not derive His subsistence from any matter; but that by His own will and counsel He has subsisted before time, and before ages, as perfect God, only begotten and unchangeable, and that before He was begotten, or created, or purposed, or established, He was not. For He was not unbegotten. We are persecuted, because we say that the Son has a beginning, but that God is without beginning. [Emphasis mine]
Both the Arian and the Nicene Christians believed that 1 Corinthians 8:6 taught that Christ is God -- certainly not a Unitarian interpretation. The Nicene Trinitarians made many arguments as to why the Son is of the essence of the Father, of one substance with the Father, but we will focus on the ones that address 1 Corinthians 8:6. The positions of the Trinitarians as they address Arianism will illustrate that they did not even consider a Unitarian understanding of 1 Corinthians 8:6 to be possible.
John Chrysostrom lived shortly after the Council of Nicaea. In his 20th Homily on Corinthians (I recommend you read the entire homily) he writes about 1 Corinthians 8:6
"yet to us there is One God, the Father." In the first instance [1 Corinthians 8:4] having expressed it without the word "Father," and said, "there is no God but one," he now adds this also, when he had utterly cast out the others. Next, he adduces what indeed is the greatest token of divinity; "of Whom are all things." For this implies also that those others are not gods. For it is said, "Let the gods who made not the heaven and the earth perish. [Jeremiah 10:11]"
So Chrysostrom has established that creatio ex nihilo is the 'is the greatest token of divinity', and he goes on to apply that to the Son:
And in regard to Christ again, we must conceive of this in like manner. For through Him the race of men was both produced out of nothing into existence, and returned from error to truth. So that as to the phrase "of Whom" it is not to be understood apart from Christ. For of Him, through Christ, were we created.
Chrysostrom understands this verse to mean that Father's action in creation (of Whom) 'is the greatest token of divinity' and can not be separated from Christ's participation (through whom) in the act of creation. This unequivocally places Christ at the same level as the Father and rules out either a Unitarian or Arian interpretation.
Let's propose that only Jesus is the lord of the Christians (and not God The Father)
There is no basis for this in the text and is not consistent with Trinitarian tradition. As if anticipating your question by some 1600 years Chrysostrom points out that if you say that 1 Corinthians 8:6 applies the word God to the Father only and exclude the Son, then you must exclude the one God, the Father from being Lord. Chrysostrom dismisses such an idea out of hand as it is obvious to him that the One God is Lord:
And this is not all, but there is another remark to make: that if you say, "Because it is said 'One God,' therefore the word God doth not apply to the Son" observe that the same holds of the Son also. For the Son also is called "One Lord," yet we do not maintain that therefore the term Lord applies to Him alone. So then, the same force which the expression "One" has, applied to the Son, it has also, applied to the Father. And as the Father is not thrust out from being the Lord, in the same sense as the Son is the Lord, because He, the Son, is spoken of as one Lord; so neither does it cast out the Son from being God, in the same sense as the Father is God, because the Father is styled One God.
Gregory of Nyssa
Gregory of Nyssa also lived shortly after Nicea. In the treatise Against Eunomius Gregory argued against an the Anomoeanism/Eunomians lead by Eunomius. They professed an extreme version of Arianism (beyond what Arius taugh) that taught that Christ was neither of the same substance (homoousia) as the Father (the Nicene position) nor a similar substance (homoiousia) as the Father (the neo/semi-Arian position) Like Chrysostrom, Gregory argues that the Son is fully separate from the creation in the same way that Father is and cites 1 Corinthians 8:6 as proof:
And let no one think it unreasonable that the creature should be set in opposition to God, but have regard to the prophets and to the Apostles. For the prophet says in the person of the Father, "My Hand made all these things" [Isaiah 66:2] meaning by "Hand," in his dark saying, the power of the Only-begotten. Now the Apostle says that all things are of the Father, and that all things are by the Son,[1 Corinthians 8:6] and the prophetic spirit in a way agrees with the Apostolic teaching, which itself also is given through the Spirit. For in the one passage, the prophet, when he says that all things are the work of the Hand of Him Who is over all, sets forth the nature of those things which have come into being in its relation to Him Who made them, while He Who made them is God over all, Who has the Hand, and by It makes all things. And again, in the other passage, the Apostle makes the same division of entities, making all things depend upon their productive cause, yet not reckoning in the number of "all things" that which produces them: so that we are hereby taught the difference of nature between the created and the uncreated, and it is shown that, in its own nature, that which makes is one thing and that which is produced is another.
Thus the Trinitarians saw 1 Corinthians 8:6 teaching that the one God, the Father, and the One Lord Jesus Christ to be united in substance because their action in creation. Though they made these arguments against the Arians, they apply equally as well, if not more so, to Unitarians. And this was not unique to the Christians of the Nicene era. The line of thought goes back well before them.
Athenagoras of Athens
Athenagoras was born around AD 133 He wrote A Plea for the Christians to the Roman emperors Marcus Aurelius and Commodus to explain and defend Christianity. In trying to prove the reasonableness of Christianity, Athenagoras [argues that the Father and the Son are one] in a manner similar to Chrysostrom and Gregory. He does not directly cite 1 Corinthians 8:6 as there would be no reason to expect the emperors to be familiar with it, but he may be alluding to it.
That we are not atheists, therefore, seeing that we acknowledge one God, uncreated, eternal, invisible, impassible, incomprehensible, illimitable, who is apprehended by the understanding only and the reason, who is encompassed by light, and beauty, and spirit, and power ineffable, by whom the universe has been created through His Logos, and set in order, and is kept in being—I have sufficiently demonstrated. I say "His Logos", for we acknowledge also a Son of God... But the Son of God is the Logos of the Father, in idea and in operation; for after the pattern of Him and by Him were all things made, the Father and the Son being one. And, the Son being in the Father and the Father in the Son, in oneness and power of spirit, the understanding and reason of the Father is the Son of God.
Athenagoras argues that Son is the Logos of the Father and one with the Father because of their action in creation. This foreshadows the later arguments of the Nicene and Post-Nicene fathers.
Justin was born around AD 100. He wrote Dialogue with Trypho a long treatise (but worth reading in full) dedicated to showing the superiority of Christianity to Judaism. It is presented as a discussion between Justin and and a Jew named Trypho. Trypho is said to have studied under Corinthus the Socratic in Argos, but now mainly lives in Corinth. In the 61st chapter, Justin writes
"I shall give you another testimony, my friends," said I, "from the Scriptures, that God begat before all creatures a Beginning, [who was] a certain rational power [proceeding] from Himself, who is called by the Holy Spirit, now the Glory of the Lord, now the Son, again Wisdom, again an Angel, then God, and then Lord and Logos; and on another occasion He calls Himself Captain, when He appeared in human form to Joshua the son of Nave (Nun). For He can be called by all those names, since He ministers to the Father’s will, and since He was begotten of the Father by an act of will; just as we see happening among ourselves: for when we give out some word, we beget the word; yet not by abscission, so as to lessen the word [which remains] in us, when we give it out: and just as we see also happening in the case of a fire, which is not lessened when it has kindled [another], but remains the same; and that which has been kindled by it likewise appears to exist by itself, not diminishing that from which it was kindled.
Now, the Trinitarian might object to the Son being called the Holy Spirit, but that is not relevant to why Justin's comments are important. Justin asserts the unity of the Logos/Son with the Father. Further, he expects a Jew educated in Greek philosophy (and living in Corinth) to be familiar with and even accept this idea. That is, both Hellenized Jews and educated Greeks of the time would be familiar with the concept of the Logos being from God, but also God, and God/Father's partner in creation. And this idea was not foreign to Hellenized Jews.
Philo of Alexandria
Philo of Alexandria, a Hellenized Jew who was roughly a contemporary of Christ, wrote of the Logos, identifying it with the image of God (Compare with Colossians 1:15 and John 1:3),
Now the image of God is the Word, by which all the world was made
I should like to have more time to expand on this, but what should be apparent by now is that not just Trinitarians but Greeks and Hellenized Jews (or at least some of them) were familiar with and accepting of the concept that the Logos was God and by whom God made all things yet without sacrificing the unity of God. Where Christianity becomes scandalous is when it identifies the Logos, which is of God and is God, with the man Jesus.
But what about Paul?
If we ignore the ignore what Trinitarians have taught on the matter and the cultural context of the time (probably not a good idea, either of them), what can we see just from the epistles? When Paul wrote,
yet for us there is one God, the Father, of whom are all things, and we for Him; and one Lord Jesus Christ, through whom are all things, and through whom we live.
did he intend separate the Father from the Son? Paul opens the 1 Corinthians placing the Father and the Son on the same level:
To the church of God which is at Corinth, to those who are sanctified in Christ Jesus, called to be saints, with all who in every place call on the name of Jesus Christ our Lord, both theirs and ours. Grace to you and peace from God our Father and the Lord Jesus Christ. [1 Corinthians 1:2-3 NKJV]
Paul places the Father and Christ side-by-side and presents them as both being the source of grace and peace. (Romans, Galatians, Ephesians ... the open the same way) He states that the saints are to call on the name of Jesus Christ. Paul, the Pharisee of Pharisees, is familiar with Joel 2:32 (He quotes it in Romans 10:13):
And it shall come to pass
That whoever calls on the name of the Lord
Shall be saved.
For in Mount Zion and in Jerusalem there shall be deliverance,
As the Lord has said,
Among the remnant whom the Lord calls.
Either Paul is being rather sloppy with the Lord's name or he intends to place his readers in the mindset that the Lord Jesus Christ is to be identified with if not identical to the Lord God of Israel.
Given this, it is difficult to believe Paul intends to separate the Father and the Son and place them on separate levels in 1 Corinthians 8:6 why describing them in parallel terms.
Paul knows that his audience is familiar with concept of the Logos. In 1 Corinthians 8:6 Paul brings up the Logos implicitly by saying "through whom are all things" and expects that they will understand the the Lord is of God and is God. And if Paul had left it as this, "the Father, of whom are all things, ... and one Logos, through whom are all things," you could read it as a Unitarian text. However, Paul goes further. He places the Lord Jesus Christ in the role of the Logos, making the man Jesus of God and God. This can not be read in a Unitarian manner.
(One BEING of God shared by Three distinct PERSONS) Is the term God used here for a being or a person?
One Being of God shared by Three Persons is not quite the Trinitarian formula. I wonder if you are are using 'being' here similar to 'human being'.
There is, per the Nicene Creed, one God, the Father. The Son and the Spirit are of the same essence/substance/being with the Father because the Father is the source of their essence/substance/being, the Son begotten and the Spirit proceeding. This could easily lead to the another question about the Monarchy of the Father.
I don't see how we can argue that the one Lord here is meant to be understood as "one YHWH" (See the idea that Paul expanded Deutronomy 6:4 in 1 Corinthians 8:6) because it is clear that Paul is drawing a line between the the lords/gods from 1 Corinthians 8:5 and we can't say that 1 Corinthians 8:5 should be understood as "many YHWH's and many gods".
It is not clear that Paul draws a line or parallel between verse 5 and 6. Rather, he could be drawing a contradiction. The pagans have many false gods and lords. We have One True God and Lord and they are one. If Paul is drawing a parallel and separating God the Father from the Lord, then he is teaching outright ditheism. In verse 5, he makes no distinction between god and lords, only saying they have many false gods and lords. Have you read Bauckham's God Crucified: Monotheism & Christology in the New Testament? He addresses this fully.