What should I read to get a good defense of Bayesianism--that isn't just pointing out difficulties with frequentism, NHST, or whatever? I understand the math, but am skeptical that it can be universally applied, due to problems with coming up with the relevant priors and likelihoods.

It's like the problem with simple deduction in philosophy. Yes, if your premises are right, valid deductions will lead you to true conclusions, but the problem is knowing whether the premises used by the old metaphysicians (or modern ones, for that matter) are true. Bayesianism fails to solve this problem for many cases (though I'm not denying that you do sometimes know the relevant probabilities).

I do definitely plan on getting my hands on a copy of Richard Carrier's new book when it comes out, so if that's currently the best defense of Bayesianism out there, I'll just wait another two months.