This is a somewhat recurrent and possibly controversial discussion, but technically the so called “Frequentist” approaches are equivalent to Bayesian inference up to the specification of priors, and therefore the former are a particular case of the latter, strictly speaking.
In practice, there end up being several differences between “Frequentists” and “Bayesians” beyond priors, but those are differences of practice, not of theoretical underpinnings. For your particular case @jsocolar points out clearly the equivalence and how to replicate it in a Bayesian framework, and in general if you understand the formal “Frequentists” statistical description it is straightforward to formulate a “Bayesian” version.