As mentioned in @spinkney’s link there is no such thing as a Jacobian for a low to high dimensional mapping. You have to first work out a bijective (one-tone) transformation with a well-defined Jacobain and then marginalize out the excess dimension in a separate step. See https://arxiv.org/abs/1010.3436 for how to do this for the simplex.
Perhaps more relevant to the application mentioned see https://betanalpha.github.io/assets/case_studies/ordinal_regression.html for an example of using domain information on integrated probabilities (in the form of a Dirichlet density function) to inform a prior model for latent parameters.