I also have a question related to such a setting. How can one compare a two class model (in terms of model fit) with a single class model when C is regressed on e.g. gender in the two class model? Can one interpret LMRT and BLRT of the two class conditional model? How can BIC be used in this situation?
I think they do. Mplus just ignores c on x for the 1-class alternative. If your model has x influencing only c, then the 1-class model says that the x's don't influence any of the y's - make sure that this is the model you are interested in.