This is an older question, but I think I can answer it.

TL:DR When controlling layers from other comps, you shouldn't use time remapping.

Explanation: Everything within the remapped comp will compare its time value to the time value of the cotaining comps. So if you set a keyframe at frame 0 in the Stage comp, it will also affect the layers within the remapped Mouth comp at frame 0. It seems you have an offset of 01:27 seconds, so if you set the keyframe at frame 0 in Stage you won't see any changes, because the Mouth comp is already ahead.