HI Tom,
Here is a discourse conversation on that exact topic. I put some code at the bottom of that post explaining how I ran models over a distribution of trees. Note that some of the model structure has recently changed (no cov_ranef anymore). Basically I made a list of trees and then looped the model over them, then used combine_models() to merge them. However, brms now has the brm_multiple() command which I personally have yet to use but I believe is created for this exact purpose. Here is the GitHub issue explaining it.
Running models on subset of the posterior distribution of phylogenetic trees is common in my field, but there may be some theoretical issues with this practice. The first discourse link I posted has some really interesting conversation in it that is worth checking out.
Here is another cool paper on this topic that takes a different view, and the authors state that as few as 50 trees from the posterior capture the uncertainty in the model from the uncertainty in the phylogenetic estimation.
Good luck!