Bivariate heritability with one of traits with a best model ADE or DCE and the other ACE with twins data

Could we run a bivariate heritability analysis using twin data when one of two traits has a best model ADE or DCE and the other ACE?

Sure, no reason why not. However, the DCE model is not recommended, because it's sort of like doing a linear regression that includes higher-order functions of a predictor without including the first-degree main effect of that predictor. The additive-genetic 'A' component represents the combined, additive/linear effects of the alleles at trait-relevant loci, whereas the dominance 'D' component represents the combined effect of within-locus interactions of alleles.

Two main phenotype seems to be more ADE for one and DCE model for the other and I compare them with other phenotypes with different best model.
One of my colleague has kindly provided to me her script but it runs only when the two traits have a best model type ACE.

I know that Ra, Rd could be null as one of my phenotype didn't have it in its best model. It is fine, but at least the model for each trait is the best.

You should post the script you want to adapt. Also, did you analyze each phenotype in a separate monophenotype analysis to decide on ADE and DCE (the latter of which you shouldn't use)? I recommend instead selecting a diphenotype model for both phenotypes. I say that because an analysis of both phenotypes utilizes information about their covariance, which is informative about the influences of A, C, D, and E.

Again, post the script you want to adapt for your purposes, and I can make some suggestions.