I can't see that dye stability should be of much concern if one uses CD-3 and does not substitute one of the other color developers. The dyes produced would be identical to those produced by the official E-6 process. If one is concerned then use Kodak's stabilizer bath rather than a home brew one. What is important is the final pH of the emulsion and neutralization of any remaining leucodyes.
I would assume that because of patent restrictions that the various films and formulas would be different. Yet Fuji film produces excellent results with the Kodak process and Kodak film in Fuji's. For example, Agfa films do not use dyes incorporated in resin beads like Kodak but rather use dyes with long side chains to prevent dye migration between color layers in the emulsion. Thus the Agfa films do not have the characteristic milky appearance that the Kodak films have before they dry. Despite different dyes the Agfa films develop correctly in E-6 chemistry.