Hi Tommy,
Did you try your models yet? I can’t really comment on Dirichlet Regression in particular, but you should be able to add the phylogenetic correlation (or covariance) matrix into basically any brms model as a group-level effect using the same notation (the data2 = list(...) part) as a gaussian or poisson regression (both are in that vignette I think). I have used the phylo matrix in ordinal models, binomial models, etc and everything works as it should! HERE is a manuscript that uses the brms phylo models in categorical regressions.
The same thing goes for the intra-specific variation part. It’s just an additional group-level effect.
One thing to consider is that if you are interested in the Pagel’s lambda value, that may not be an option in non-gaussian regression (or regressions that don’t include a sigma variable). There is some discussion of that here in this post.
Good luck!
[edit: spelling]