Have you thought about using latent multivariate normals to specify this correlation across categories/properties?
Its straightforward to convert their output to the simplex domain (see here), while at the same time using the tools that already exist for modelling covariance structures.
Otherwise it seems like you are looking for more generalized versions of the Dirichlet, which have also been described in the literature. Not sure if there’s something taylor made for your situation, as your explanation of the problem is a bit vague (no offense!).