JonJ: You say there is knowledge beyond the reach of science. OK, what is it? Can you give us one tiny crumb of incontrovertible fact sourced from any method other than science?
Science gets it wrong at least half the time. But religion gets it wrong all the time, because it never checks.
VR: I don't think religion never checks. I'm not a fideist. I think that there are ways of checking my beliefs. There are possible arrangements of evidence that would make me doubt my religion. What happens from there is anybody's guess, but I don't follow Craig in saying that the Holy Spirit gives me such reassurance that, given any possible change in the evidence or my evaluation of the evidence. I go with C. S. Lewis's "I am not asking anyone to accept Christianity if his best reasoning tells him that the weight of evidence is against it."
I think you have to distinguish between narrowly scientific reasoning and broadly scientific reasoning. Narrowly scientific reasoning is the kind of reasoning accepted in various scientific disciplines. It has some common overall themes, but differs somewhat from disciple to disciples. Should sociology try to be just like physics? Probably not, but of course there are going to be some similarities.
If we are speaking of broadly scientific reasoning, then I can't see any reason why the reasoning one engages in in determining religious beliefs can't be broadly scientific. I believe in Bayesian conditionalization, and I do believe that what I believe religiously can be either confirmed or disconfirmed by evidence. There is something I call the "vacuity argument" which says that "supernatural" (and here the idea of the supernatural has to be clarified, because according to some ways of defining the term I don't think even God is supernatural, and I see no good reason in theory why God couldn't be a theoretical entity in a scientific explanation), explanations are excluded because they can just be stuck in anywhere, and are therefore vacuous. But I have never found this argument persuasive in the least.