An alternative approach is to not use a particular copula or to assume the marginals follow a specific distribution. Rather, there is a semi-parametric copula approach based upon the rank likelihood. In this case, “Any continuous multivariate distribution can be used to form a copula model via
an inverse-CDF transformation” (p. 270). Then, similar to a probit model, the ranks are used to sample the underlying latent data. The nice aspects are that there is no need to sample the cutpoints for ordinal data and this works for “mixed data”. And, in fact, many are now using this approach in lieu of a multivariate probit model. See the R packages BGGM and BDgraph.
Here is the paper.
“EXTENDING THE RANK LIKELIHOOD FOR SEMIPARAMETRIC
COPULA ESTIMATION”