Introduction: A full explanation for the biblical basis of the doctrine of the Trinity can be found here: https://irr.org/biblical-basis-of-doctrine-of-trinity
Because the OP asks for the biblical basis for claiming that belief in the Trinity is necessary for salvation, I will not attempt to prove the biblical basis for the Trinity but simply refer the reader to the article above by the Institute for Religious Research (May 2011). It is a lengthy document that cites about 1,000 references from over 300 different chapters of the Bible, including references from all 27 books of the New Testament.
Parts I and II of the study begin by establishing the fact that there is only one God, the Lord. Part III shows from the Bible that the Father of our Lord Jesus Christ is God; Part IV that the Son, Jesus Christ, is God; Part V that the Holy Spirit is God. Part VI explains how the Father, Son, and Holy Spirit are each distinct from the other two and Part VII summarises how the Bible teaches the Trinity. For a quick review of the importance of the Trinity, read Part VIII: https://bib.irr.org/biblical-basis-of-doctrine-of-trinity-part-viii-what-difference-does-trinity-make
The objective of what follows is to establish the biblical basis for the necessity of accepting the Trinity doctrine. If we reject the deity of Jesus, the Son, and the deity of the Holy Spirit in their roles within God’s eternal plan of salvation, then how can we expect to be saved?
With regard to the role of Jesus Christ in salvation, here is a short extract from Part IV, section C 2, which shows that the title of Saviour, which belongs in the ultimate sense only to God, is ascribed to Jesus as the source of salvation. Remember there is only one true God who is called Saviour – yet this title is held by Jesus.
Today in the town of David a Saviour has been born to you; he is Christ the Lord (Luke 2:12)
With regard to the Samaritans who became believers in Jesus, “now we have heard for ourselves, and we know that this man really is the Saviour of the world” (John 4:42).
Not only is Jesus our Saviour, Jesus is also Lord:
and you will receive a rich welcome into the eternal kingdom of our Lord and Saviour Jesus Christ (2 Peter 1:11).
But grow in the grace and knowledge of our Lord and Saviour Jesus Christ. To him be glory both now and for ever! Amen” (2 Peter 3:18).
Yet 1 Timothy 4:10 declares that the first Christians put their hope in the living God, who is the Saviour of all men. How can it be then, that Jesus is the source of salvation?
he became the source of eternal salvation for all who obey him (Hebrews 5:9).
Part IV, section 5 is also worthy of your consideration with regard to the role Jesus plays in salvation. It can be summed up in John 5:21:
For just as the Father raises the dead and gives them life, even so the Son gives life to whom he is pleased to give it. Moreover, the Father judges no-one, but has entrusted all judgment to the Son, that all may honour the Son just as they honour the Father. He who does not honour the Son does not honour the Father, who sent him.
With regard to the role of the Holy Spirit in salvation, Romans 8:1-27 spells out His role in no uncertain terms. It is the Spirit of life that sets us free from the law of sin and death. It is the power of the indwelling Holy Spirit that transforms the believer. The believer is controlled by the Spirit:
And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you (Romans 8:11).
Acts 5:3-4 clearly establishes the biblical truth that to lie to the Holy Spirit is to lie to God. 2 Corinthians 3:17-18 confirms that the Holy Spirit is the Lord:
Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. And we, who with unveiled faces all reflect the Lord’s glory, are being transformed into his likeness with ever-increasing glory, which comes from the Lord, who is the Spirit.
The Holy Spirit is Eternal (Hebrews 4:19), was involved in Creation (Genesis 1:2; Psalm 104:30; Job 33:4); was responsible for the Incarnation (Matthew 1:18, 20; Luke 1:35); Intercedes for Christians with the Father (Romans 8:26).
The Holy Spirit is sent by Christ to convict of sin, speaks not on his own but on behalf of Christ, and always glorifies Christ. Christians must be filled with the Spirit (Ephesians 5:18). His role in our salvation is described in John 16:7-14. This is no mere source of power or energy – this is God at work in the process of salvation.
Conclusion: The OP seems to have no problem in accepting the incarnation, that God dwelt with us in human form as Jesus, yet fails to understand the role of the Holy Spirit in salvation. One way of looking at it is to realise that just as Jesus was God with us back then, so too the Holy Spirit is God with us now. Father, Son and Holy Spirit are inextricably linked in the work of salvation. Without acknowledging who Jesus Christ REALLY is and without understanding who the Holy Spirit REALLY is, believers are in spiritual darkness.
Sometimes it takes a long time before the scales are removed and the believer finally understands the eternal relationship between Father, Son and Holy Spirit. But for those who cling to the notion that before Jesus came to earth he was either created or born as a mighty spirit creature, and who think the Holy Spirit is just some sort of impersonal source of energy from God, then they are spiritually blind.
Only by understanding the roles all three play within the Godhead can Christians give worship, praise and glory to the one true God, to worship Him in spirit and in truth. That is why the Athanasian Creed declares, without apology or excuse, that “we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence. . . . He therefore that will be saved, let him thus think of the Trinity.”
The full Athanasian Creed can be found here: https://www.crcna.org/welcome/beliefs/creeds/athanasian-creed
Caveat: The Athanasian Creed demands belief in all of its tenets for salvation. While in agreement with the tenets, I do not believe that all of them are mandatory for salvation.
Explanation needed: When I first came to saving faith in Christ Jesus I did not believe in the Trinity doctrine. In fact, I was anti-trinitarian. But that did not prevent God from responding to my heart-felt repentance and my plea to be forgiven for my sins. The Holy Spirit confirmed to me that I had been adopted into God's family. The spiritual heart transplant happens before the heavy-duty theological stuff can get into your head. God never expects his children to run before they can walk, spiritually speaking. It took time for the reality of who Christ Jesus and the Holy Spirit REALLY are to sink in. But God was patient with me (as He is with all who sincerely repent and step out in faith), and the message got through later.
Having said that, I now realise that to deny the eternal relationship between Father, Son and Holy Spirit results in a wrong understanding of the plan of salvation. It's not enough to say you believe in the Incarnation yet exclude the role of the Holy Spirit. There is only one true God who is composed of Father, Son and Holy Spirit, and that is why "we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence."