I was suggesting replacing the first Dirichlet (v_t) itself by another prior on the simplex. The alpha parameter need not come from a simplex but it’s a common parametrization I’ve seen: alpha = simplex * concentration.
If I read you correctly you are suggesting dropping the first Dirichlet/ simplex (v_t) altogether and parametrize the observed Dirichlet alpha directly with a positive process? That would be my first approach as well but I don’t know if @Bryan_Palmer had a reason to start with that model.