There is an excellent paper about using copulas on count data:
A PRIMER ON COPULAS FOR COUNT DATA
BY CHRISTIAN GENEST AND JOHANNA NESLEHOVA
Advantage of copulas are to model tail dependencies, multinormal gaussian only consider
linear dependencies between random variables.
Some work had only been done in Stan by Ben Lampert, it’s based upon the Poisson
distribution, but can easily enhanced to other discrete distributions.
Copulas are one way to go - a very flexible. This comes to a price though. Technically demanding,
but solvable. Then also there’s the need of more data or stronger priors.
One has to consider if not multivariate extensions, eg. Laguerre Polynom(s) already may be
suitable. Then it comes to the point where we have to analyse your data,
happily, if not, we already have had been replaced by some sophisticated deep learning
whats’o’ever.
Your question is to vague to give an answer. (By no means I want to offend anybody)
Just my 2 cents.