The non-Christian is ultimately the skeptic because he is denying a central tenet to the laws of logic that he already presupposes; perhaps we could say he is denying a version of the principle of sufficient reason, and therefore is under the burden of proof of why the PSR doesn’t apply to his assumptions.
But let's be clear here: PSR is not germaine to the question of whether or not the unbeliever is warranted in using reason or whether the reasoning process is de facto borrowed from the Christian worldview. The use of reason is self-evidently warranted.
As for PSR, there has to be a point where the chain ends---our disagreement is over where. The unbeliever is simply arguing for a multiplicity of places---it's a messy position, but not (from my understanding) self-contradictory.
Do reasons exist?
Ok, here's where you and I are talking past each other: when I talk about a reason for a belief I am simply speaking of the reason why I believe X. You, however, are speaking of the reason why X is the case.
Therefore, an Atom holds together for the reason of electromagnetic force; but this reason exists contingently (i.e., it is a contingent reason).
The term doesn't make sense---that's my argument. The reason for X is Y and Y is a contingent fact. But that doesn't mean that it's relation to X is contingent---its relation to X may be necessary. It may be the de facto reason for the truth of X, but its relation to X is an open question apart from that.
but this reason exists contingently (i.e., it is a contingent reason).
No, I don't believe that reasons exist---I'm not a Platonist or a Scotist.
Forgive me if this is too simple a reduction but is this the thrust of your challenge to transcendental arguments, that an unbeliever doesn't have to provide a basis for their presuppositions? Surely if certain presuppositions render other aspects of their world-view contradictory and inconsistent, then they would have to give an account for these presupps?
Part of the issue is this:
a) Presuppositions are the foundation in the sense that they are informing principles that shape the way we think. The very term Presupposition implies that it is a basic component of one's epistemic structure.
b) They aren't propositional at all, or rather their propositional forms are merely manifestations of underlying attitudes and precommitments, methodologies, and predispositions. How would one give an account of a predisposition?