Is there Biblical support for the concept that 'Whenever a man lies with a woman...a transcendental relation is set up between them'?

This particular quote from C.S. Lewis comes from Letter XVIII of The Screwtape Letters. Screwtape, a "senior undersecretary" devil, is writing to his nephew, Wormwood, on the subject of sex and sexual temptation. After talking about the concept of "love", and how it is "gratuitously associated" with sexual desire (and how hard that makes the devils' job), Screwtape continues:

Now comes the joke. The Enemy described a married couple as "one flesh". He did not say "a happily married couple" or "a couple who married because they were in love", but you can make the humans ignore that. You can also make them forget that the man they call Paul did not confine it to married couples. Mere copulation, for him, makes "one flesh". You can thus get the humans to accept as rhetorical eulogies of "being in love" what were in fact plain descriptions of the real significance of sexual intercourse. The truth is that wherever a man lies with a woman, there, whether they like it or not, a transcendental relation is set up between them which must be eternally enjoyed or eternally endured.

From the true statement that this transcendental relation was intended to produce, and, if obediently entered into, too often will produce, affection and the family, humans can be made to infer the false belief that the blend of affection, fear, and desire which they call "being in love" is the only thing that makes marriage either happy or holy.

"Mere copulation, for him, makes 'one flesh'" is doubtless, as others have pointed out, a reference to 1 Corinthians 6:16:

Do you not know that anyone who joins himself to a prostitute becomes one body with her? For "the two," it says, "will become one flesh."

Paul is not talking about marriage yet; he does that in chapter 7. Here he is simply talking about sexuality and sexual activity in general, yet he pulls in the very same text that is usually used to speak about marriage. This is why Screwtape says that "mere copulation" makes "one flesh".

As far as "eternally enjoyed or eternally endured": I don't think that Lewis (or Screwtape) intended to say, contrary to Luke 20:35, that "a man [who] lies with a woman" is married to her eternally (i.e. even in Heaven). That doesn't seem to be what he's talking about; he's talking about how the relation between the man and the woman which is caused by the act of sexual intercourse can be built into a happy, holy marriage, or can be used to break down a person's concept of love and marriage:

[If humans are led to misunderstand the real significance of sexual intercourse] humans who have not the gift of continence can be deterred from seeking marriage as a solution because they do not find themselves "in love", and, thanks to us, the idea of marrying with any other motive seems to them low and cynical. Yes, they think that. They regard the intention of loyalty to a partnership for mutual help, for the preservation of chastity, and for the transmission of life, as something lower than a storm of emotion. ... [Furthermore] any sexual infatuation whatever, so long as it intends marriage, will be regarded as "love", and "love" will be held to excuse a man from all the guilt, and to protect him from all the consequences, of marrying a heathen, a fool, or a wanton.

This, in Screwtape's and Lewis's eyes, is the mistake that will have to be "eternally endured"—because it can lead to a situation in which it's difficult or impossible to live as a Christian. Screwtape gets into this in more depth in letter XIX.

We do know that something does happen.

The following appears to be the relevant biblical passage to your question, with attention being drawn to the line in bold:

1 Cor 6:12-15 (RSVCE)
Glorify God in Body and Spirit
12 “All things are lawful for me,” but not all things are helpful. “All things are lawful for me,”[a] but I will not be enslaved by anything. 13 “Food is meant for the stomach and the stomach for food”—and God will destroy both one and the other. The body is not meant for immorality,[b] but for the Lord, and the Lord for the body. 14 And God raised the Lord and will also raise us up by his power. 15 Do you not know that your bodies are members of Christ? Shall I therefore take the members of Christ and make them members of a prostitute? Never!

Footnotes:
a. 6.12 This saying is possibly an exaggeration of the freedom from the Mosaic law that Christians enjoyed. The saying has been applied to sinful practices, as is clear from the following verses.
b. 6.13, 18 immorality: i.e., sexual immorality.

For an experiential and modern views please see:

• Letter to a Young Girl | Alice Von Hildebrand
• The section The Science Behind Sex
• The paragraph that starts Abstinence educators use a simple demonstration to open discussions of sex with adolescents.

eternally enjoyed or eternally endured

CS Lewis himself answers as follows:

“Son,'he said,' ye cannot in your present state understand eternity...That is what mortals misunderstand. They say of some temporal suffering, "No future bliss can make up for it," not knowing that Heaven, once attained, will work backwards and turn even that agony into a glory. And of some sinful pleasure they say "Let me have but this and I'll take the consequences": little dreaming how damnation will spread back and back into their past and contaminate the pleasure of the sin. Both processes begin even before death. The good man's past begins to change so that his forgiven sins and remembered sorrows take on the quality of Heaven: the bad man's past already conforms to his badness and is filled only with dreariness. And that is why...the Blessed will say "We have never lived anywhere except in Heaven, : and the Lost, "We were always in Hell." And both will speak truly.” ― C.S. Lewis, The Great Divorce

There's a very common teaching on what's called "soul ties".

The idea is that there are various things that can tie two human souls together. One of those is sex. The Biblical support for this is 1 Corinthians 6:16

6 Do you not know that he who unites himself with a prostitute is one with her in body? For it is said, “The two will become one flesh.”

How this equates to a "soul tie" is unclear to me, but to the adherents, it seems to implicitly make sense.

A sample of the teaching can be found here.

Excerpt:

Brief Biblical look at soul ties

1 Corinthians 6:16, "What? know ye not that he which is joined to an harlot is one body? for two, saith he, shall be one flesh."

Obviously the two didn't get married, but something spiritual "happened" when they were joined physically in the act of sex. They were "joined"; their souls were "tied." Often there are men who have an adultery with a prostitute (one night stand), and afterwards, even years later, are still thinking about that girl! That is because a soul tie has been created.

There's another article here describing the same thing, going deeper into it and attempting to show that there are ties to body, soul, and spirit.

Again, I don't see these as a natural link, personally, and whether this is strong enough of a "Biblical basis" for any given individual is a personal opinion, but for those that teach and believe in the "soul ties" phenomenon, this is the basis given.