what you're doing here is conflating Bayes' theorem (which is about probability, and which is a matter of logic, and which is correct) with Bayesian epistemology (the application of Bayes' theorem to epistemological problems, rather than to the math behind betting).

That's because to a Bayesian, these things are the same thing. Epistemology is all about probability - and visa versa. Bayes's theorem includes induction and confirmation. You can't accept Bayes's theorem and reject induction without crazy inconsistency - and Bayes's theorem is just the math of probability theory.