I am interested to know if this is possible, I would like to do this for example so that I can export the lip sync to a 3d app and use as a animated texture. The movement, head tracking would destroy this illusion though.
I've recorded just the mouth by opening the puppet in Photoshop and making all layers except the mouth invisible. After saving the file Ch sees only the mouth, like the Cheshire cat. This may also work for the eyes, but I haven't tried that.
You might also be able to just set the Face behavior's Head Position/Scale/Tilt Strength and Parallax Strength to 0.