You are asking to be pointed towards church fathers that believe a similar mindset as your line of reasoning. This is... unlikely.
I'm eastern Orthodox, so I totally get your request for a reference to church fathers. :)
However, the doctrine of the Trinity was declared in 381 in the Nicene-Constaninopolitan Creed. This is when the Trinity that was understand today was officially confirmed.
Keep in mind that the old Church generally did not make declarations unless something wrong was happening. Usually someone was teaching a heresy. This puts any ancient source (church fathers) in a dubious position as it would likely have to be one of the heretics and not a church father.
Another thing to consider here is that... We cannot rationalize our minds to God.
Let us look to some of the church fathers and then the Scriptures themselves.
Do not slight the discourse of the sages,
but turn to their maxims,
because from them you will learn discipline
and how to serve the great. (Wisdom of Sirach 8:8)
(Orthodox) Christians believe in God the Trinity - Father, Son and Holy Spirit. The Trinity is not three gods, but one God in three Hypostases, in three personal beings. What mathematics and logic consider an absurdity constitutes the cornerstone of our faith, namely that 1=3 and 3=1. Christians participate in the trinitarian Godhead not through logic but through repentance, that is, through a complete change and renewal of the mind, heart and feelings (the Greek word for 'repentance' - metanoia - literally means 'change of mind'). To touch upon the mystery of the Holy Trinity is impossible unless the mind changes from a rational way of thinking and becomes illumined by divine grace. (Diocese of Sourozh, Russian Orthodox)
God is unoriginate, unending, eternal, constant, uncreated, unchanging, unalterable, simple, incomplex, bodiless, invisible, intangible, indescribable, without bounds, inaccessible to the mind, uncontainable, incomprehensible, good, righteous, that Creator of all creatures, the almighty Pantocrator --(St. John Damascene, "Chrysorrhoas" 700ish AD).
Far removed is the Father of all from those things which operate among men, the affections and passions. He is simple, not composed of parts, without structure, altogether like and equal to himself alone. He is all mind, all spirit, all thought, all intelligence, all reason (Hieromartyr Irenaeus, Against Heresies 2:13:3 [A.D. 189] Pre-Nicene Council).
“No one can rightly express him wholly. For on account of his greatness he is ranked as the All, and is the Father of the universe. Nor are any parts to be predicated of him. For the One is indivisible; wherefore also it is infinite, not considered with reference to inscrutability, but with reference to its being without dimensions, and not having a limit. And therefore it is without form” (St. Clement of Alexandria, Miscellanies 5:12 [A.D. 208]) Pre-Nicene Council.
“God, however, being without parts, is Father of the Son without division and without being acted upon. For neither is there an effluence from that which is incorporeal, nor is there anything flowering into him from without, as in the case of men. Being simple in nature, he is Father of one only Son” (St. Athanasius (the Great) of Alexandria, Letter on the Council of Nicaea 11 [A.D. 350]).
“God is of a simple nature, not conjoined nor composite. Nothing can be added to him. He has in his nature only what is divine, filling up everything, never himself confused with anything, penetrating everything, never himself being penetrated, everywhere complete, and present at the same time in heaven, on earth, and in the farthest reaches of the sea, incomprehensible to the sight” (St. Ambrose of Milan, The Faith 1:16:106 [A.D. 379]).
At this point I could continue to list more and more Church fathers that firmly support the fact that God is unchanging. But let's go back even further...
“But you [God] remain the same, and your years will never end” (Psalm 102:27).
“Your word, LORD, is eternal; it stands firm in the heavens” (Psalm 119:89).
“The grass withers and the flowers fall, but the word of our God endures forever” (Isaiah 40:8).
“I the Lord do not change” (Malachi 3:6).
“’I am the Alpha and the Omega—the beginning and the end,’ says the Lord God. ‘I am the one who is, who always was, and who is still to come—the Almighty One.’” Revelation 1:8
Every good gift and every perfect gift is from above, coming down from the Father of lights, with whom there is no variation or shadow due to change. (James 1:17)
Jesus Christ is the same yesterday and today and forever. (Hebrews 13:8)
God did not "split up", because God is unchanging.
There are no church fathers that agree with you, there are heretics who might. (Arians or Gnostics perhaps)
Peace be with you.