I’m probably so far down the copula rabbit hole that (potentially better) alternatives are a distant hope… sunk costs and all that 
I think copulas are quite nice in the case that you want to allow the marginals to correlate in a pooled fashion regardless of the sub-models on each marginal - my intention is that this gains stability and flexibility in the design of the sub-models. Also to more easily use non-Gaussian copulas.
The alternative would be to require several features to form the sub-models of both marginals, and to correlate the sub-model coefficients, I think this would be a good example: McElreath, 2014 where he uses an MvN to correlate hierarchical hyperparams. I’m not sure if/how one could achieve this with a non-Gaussian copula…