Yes, putting a dirchlet prior on tau_prop looks mostly sensible, but is probably not directly interpretable. If you really want an interpretable prior, then I think you could put a dirichlet prior on the implied distribution of responses when mu is at some reference value (probably zero). That would require some transformations and Jacobian adjustments similar to the prior choice discussed at Mike Betancourt’s Ordinal Regression case study (but note that in the case study, none of the thresholds are fixed - which introduces problems if some of the extreme categories are rare in the data, but that shouldn’t be a concern for your model).
This usually requires a bit more detailed investigation. The discussion in Mike’s case study on hierarchical models describes when one would expect centered vs. non-centered parametrization to work better.
Additionally, I have recently encountered a couple situations where there was a strong negative correlation between random intercepts and the overall intercept (thresholds in your case), which then posed sampling problems. (i.e. the model allows all the random intercepts to go up while the main intercept goes down). You can investigate pairs plots for the relevant parameters to see if this is the case. I don’t think I understand this type of problems well at this moment, but enforcing a sum-to-zero constraint on the random intercepts tended to help quite a bit (but somehwat changes the interpretation of the coefficients!).
Probably more than you need to know about implementing sum-to-zero constraint: Test: Soft vs Hard sum-to-zero constrain + choosing the right prior for soft constrain
TLDR: either use sum(x) ~ normal(0, <small_number> * <num_lements>) or the code in this answer in the aformentioned thread + the one below (which is in my experience slightly superior). For more details read the whole topic.
Hope that helps at least a bit.