That’s weird with both meshes parented to the same armature it should transfer with the normal import settings, check they both are parented to the same rig (have the armature modifier with the same name on the object space), select first your meshes and last the armature (don’t know if it actually matters, I just do it like that) and normal .fbx export.
For the .fbx export, I use to check just the armature and the mesh on the include section set smoothing to face on geometry section, and check only deform bones and uncheck add leaf bones on the armature section.
I hope it helps you, tell me how it goes.