Rigging

I am pretty sure you just create a puppet with the mouth and no other layers. CH does not understand what the artwork looks like (much) - it just moves around whatever it is given. So I think you could certainly create a lipsync of just mouth and lips, export a video with transparency (or dynamic link etc), and overlap that on another image or video in After Effects.

There is even a sample mouth pack for download you could experiment with. Just load it as your puppet and see what happens!