I still don’t quite get what you are looking for I think. Because
az.plot_pair(idata,kind='kde',var_names=['w_zerosum__'])
comes out pretty similar other than scales
Well-behaving (False, False)
vs
Seemingly overdispersed (True, True)
Also - when you say hard to reason about all the permutations, what do you mean?
The model itself is a rather straightforward multinomial glm, just one with and the other without intercept.