Ultra-cold dipolar mixtures in component-dependent optical potentials constitute an interesting platform for the study of the interplay between intra- and inter-component anisotropic long-range interactions. We study the particular case of binary dipolar mixtures placed in separated bilayers which are displaced in an anti-magic wavelength configuration. Using a combination of second-order perturbation theory and cluster-Gutzwiller calculations, we unveil a rich landscape of possible crystalline phases for the two components, showing that, interestingly, inter-site hopping may result, via super-exchange, in solid-into-solid transitions between different crystalline phases. These crystalline phases and the corresponding transitions can be experimentally realized using e.g. lanthanide mixtures in optical lattices.