Okay so you are trying to use posterior parameters from a separate global model as a fixed prior to your local models?
Exactly. I believe that is similar to Two-stage Bayesian regression enforcing a fixed distribution (not Just Hierarchical regression) - #18 by Ray_Kruger. Including the maths.
Dumb question: why must they be fixed? Don’t your local models give you more information about your global parameters?
That’s certainly not a dumb question. A bad reason, but a practical one, is that the sampling fails with more than 20 or so stations. That likely has to do with the model formulation. Possibly because the stations are correlated in some ways, and accounting for the correlation is tricky. Dealing with the stations separately bypass that difficulty.
I am working towards having one global model, but even then, I find it insightful to have a decoupled model as point of comparison.