Since no one else answered, I will give it a try.
Maybe I am missing something, but what would be problematic with making the prior sd of \beta_B be the sd of the posterior for \beta_B from the previous study? In a more general language, you are trying to approximate the actual posterior density (given via the samples from Stan) with an analytic density (normal in this case). This would IMHO correspond to normal Bayesian updating which has good theoretical claims to being the best you can do.
Hope that helps!