I think what you need is explained here, and without the use of copulas: https://www.r-bloggers.com/easily-generate-correlated-variables-from-any-distribution-without-copulas/

(I cannot still add comments, so I post it as an answer)

I think what you need is explained here, and without the use of copulas: https://www.r-bloggers.com/easily-generate-correlated-variables-from-any-distribution-without-copulas/

In the words of the author of the linked article (Francis Smart), the proposed method is:

1. Draw any number of variables from a joint normal distribution.

2. Apply the univariate normal CDF of variables to derive probabilities for each variable.

3. Finally, apply the inverse CDF of any distribution to simulate draws from that distribution.

The result is that the final variables are correlated in a similar manner to that of the original variables. This is because the rank order of the variables is maintained and thus correlations are approximately the same though not exact.

Hope it helps.