I'll probably try to estimate the parameters θ from x in a way that has the best expected success rate across all possible data sets M would generate.

Isn't this a Bayesian method? The phrase "best expected" seems like a decent hint that it is. A Frequentist method would try and guaranty something like (minimax) how good your estimate is if θ changes across trials in a maximally inconvenient way.

I know you said not to claim that you were a Bayesian all along, but it seems to me that calculating a risk that depends on θ is just plain the wrong thing to do.