In the case that belief in God (specifically the Christian God) could decisively be demonstrated to be irrational (via some combination of logic and evidence), would this commit Kierkegaard's Knight of Faith to embrace irrationality?
I think the answer to this question is going to be that you're going to have to do a lot of work to explain what you mean by "demonstrated to be irrational." Kierkegaard writes a good deal about proof for the existence of God.
Three highly relevant texts are going to be:
Philosophical Fragments by Johannes Climacus
Concluding Unscientific Postscript to the Philosophical Fragments by Johannes Climacus
Practice in Christianity by Johannes Anti-Climacus
Johannes Climacus writing in the wake of Hegel articulates the view that "proof" is the wrong notion to be looking for at all. Jan Evans explains in her book on Miguel de Unamo as follows:
Johannes Climacus says that it is ridiculous to attempt to prove the existence of someone who is present to the exister and it is actually an affront (CUP 545). In Philosophical Fragments, he makes fun of anyone who would want to prove God's existence saying, "Therefore, anyone who wants to demonstrate the existence of God ... proves something else instead, at times something that perhaps did not even need demonstrating, and in any case never anything better. For the fool says in his heart that there is no God, but he who says in his heart to others: Just wait a little and I shall demonstrate it -- ah, what a rare wise man is he" (PF 43) (Evans 72).
In other words, Kierkegaard (at least in the pseudonym Climacus) is not looking for any proof. The word "demonstrate" here reflects a particular Hegelian idea of demonstration, but Kierkegaard is no friend to the one who thinks they have an objective proof for the existence of God.
Thus, the existence of God is not the sort of thing that could be
decisively demonstrated in any direction. In fact, that's a misunderstanding of what our relation is to God for Kierkegaard.
This is clear if you look at the Anti-Climacus works as well. There, we see that Kierkegaard argues that our relation to Jesus is no different than the disciples, because while we can complain we haven't met Jesus, they can complain that the Jesus they met is not someone who would appear to be God. Thus, both must relate in faith, rather than in objective proof. Because in objective proof faith is lost.
There might be a passage where Kierkegaard addresses when someone thinks they've found an objective proof God does not exist, but I can't recall it off the top of my head. I'm moderately confident he would either (a) temporarily accede and then string out a proof by ironic contradiction or (b) immediately point to the oddity (on his view) of trying to use proof in this way (which is more than he thinks it can accomplish).
Those are two other pseudonyms. It should be noted that within the maze of pseudonyms, Anti-Climacus is "Christian" but Johannes Climacus and de Silentio are both supposedly non-Christians. Returning to de Silentio, part of the point is that there's no such thing as
definitive proof one way or the other for Kgd (thus, he's considered a father of existentialism). Instead, there's "objective" (in the sense of commonly believed and works according to someone's logic) and "subjective" (originating in the subject). Neither objective proofs of God's existence nor objective proofs of God's non-existence should way the knight of faith from his subjective belief, because all "objective" proofs are subject to further revision.
tl;dr - since Kierkegaard doesn't think definitive objective proofs are possible, there's not much in the Kierkegaard secondary literature to address what de Silentio would do if such a proof existed.