Quote - UV mapper didn't work...
I loaded the grouped model, exported UV's (it does export the group info)
When I loaded the morphed mesh and tried to reload UV's, it said incompatible facet structure.

What you can try is saving the grouped OBJ from your modeller before saving the UVs in the hope that it changes the face structure to be the same as the morph OBJs.
If that doesn't work, I'd need to know more details like which modeller your using, and the steps you've used to produce the various OBJ files.