What are some Bayesian ways to see if two empirical distributions come from the same data-generating process? Are any of the Bayesian alternatives implemented in Matlab/R/Python?
Given that the two-sample Kolmogorov-Smirnov test is a nonparametric procedure, you have to dig into the area of Bayesian nonparametrics to find Bayesian analogous tools.
Here is a paper where you can start:
I warn you that nonparametric Bayesian theory is not easy to digest.