Yes the CODA paper is what I meant.

The right way out is to have a "weird" prior that mirrors frequentist behavior. Which, as the authors point out, is perfectly fine, but why bother? By the way Bayes can't use Horvitz-Thompson directly because it's not a likelihood based estimator, I think you have to somehow bake the entire thing into the prior.

The insight that lets you structure your B setup properly here is sort of coming from "the outside the problem," too.