Not sure how helpful it is but I’ve played around a lot with latent dynamic factor models lately so I think I can offer some advice.
First, it is often necessary to impose a specific set of constraints on the factor loadings to avoid identifiability issues. Of course these constraints can mean that different chains have different solutions, but there is a convenient trick one can use in generated quantities to sign correct the loadings / factors (see a detailed discussion of both issues in this Discourse post).
Second, if you hope to eventually change from AR to VAR dynamics, you will certainly find some explosive solutions due to the fact that, proportionally, the vast majority of possible parameter space typically leads to nonstationarity in VARs (see this paper by Sarah Heaps for more details). Sarah provides some useful functions that can be implemented in Stan to enforce stationarity of the VAR while allowing for more informed priors to be used, so I’d recommend you have a look there if VARs are your goal.