Scientism of the kind that Dawkins espouses is indeed in its death throes. Yes, it's prevalent in the hard sciences, but about nowhere else. Even in analytic philosophy, scientistic thinking is seen as something of an embarrassment.It would be nice if Dawkins's approach were it in its death throes, but I seriously doubt that scientism is and there's little evidence to support that. Most mainstream scientists are naturalistic and scientistic and nothing that I know suggests a change in that any time soon.
Here's my concern: in theory, I see how a transcendental argument could work. However, the claim it makes will always come across as arrogance for this reason: the claim made is such that the burden of proof required to establish it would take several lifetimes. The impossibility of the contrary may be established either by a direct argument from logical necessity, or an indirect argument from a) logical coherence (and consistency with reality), which always ends up being an argument from lack of unanswerable objections b) deconstruction of all other possible worldviews.Only the Christian worldview furnishes the necessary preconditions for intelligibility and thus, properly, not only deserves a place at the table but is the only one that does. This sounds, and is, audacious (and I realize that you, P.F., don't buy it), but it's not the audacity of the unbeliever rejecting God and His Word; rather it is the conviction and assurance of God's child humbly bowing before Him and His revelation, without which we cannot make sense of the world and with which we can joyfully serve our God and king.
In theory, the contrary is impossible, but one is forced, as a creature with limits on time, energy, and reasoning abilities, to find arguments that are less cumbersome. If you can prove that Christianity is, indeed, the only possible academically-respectable view, then do so. But if you're going to make this claim in the academy, then a demonstration is necessary.
My problem with it is that it's a burden of proof that is impossible to actually fulfill. It's a claim so strong as to be not demonstrable.
The other problem, of course, is that this isn't how anyone comes to faith. People come to faith because the Holy Spirit illumines their hearts, reveals Christ, and regenerates them---that's the epistemic basis for our knowledge of God. A transcendental argument wouldn't convince me out of any position, only to nuance it---and that would only be if I admitted that such an argument was both interesting (difficult) and necessary (more difficult).