I don’t think I’d call current medical stat practices “frequentist”; they are an incoherent mix of Neyman-Pearson behavioral decision theory and Fisher’s evidential theory.
As has been pointed out by @Sander_Greenland in a number of threads and other venues, admissible frequentist techniques (in the sense of not being dominated by another method) are in the complete class of Bayes rules. There is nothing wrong (and much to be gained) with taking a frequentist estimate, asserting a posterior, and computing the range of priors compatible with those results. @Robert_Matthews and Leonhard Held have adapted this technique from IJ Good and gave it a better name: Bayesian Analysis of Credibility.
Links to papers are in this thread:
My only adaptation is to extend this model with concepts from Game Theoretic Probability that Glen Shafer and Vladimir Vovk have explored. I would compute Skeptic and Advocate priors that gives a range of plausible prior probability distributions, leading us into the realm of imprecise probability and Robust Bayesian Analysis.
This has direct relevance to the design of experiments, in that we derive the experiment that narrows this region of probabilities, giving explicit information interpretations to the result as well.
David R. Bickel has done some similar work along these lines. In his model, a parameter K, on the scale 0-1 controls how much caution to place on the model, with numbers closer to 1 being more conservative (and closer to the frequentist procedure). I’d reverse the scale and re-interpret this a what Kelly fraction someone would be willing to bet at.