Hi,
I think the closest thing I’ve seen is the multivariate probit model (discussed e.g. here: Feedback request: Multivariate Probit Regression with GP) where some clever tricks are done to integrate out nuisance parameters that represent the latent multivariate normal.
I am not really sure I understand how multinomial sampling enters the picture as beta-binomial has binomial sampling. Could you be more specific about the generative process you have in mind?
M.