Are you exporting to FBX format? Might help to make a control rig, a simple fk rig that's driven by the Mixamo and custom rig, which the mesh is then bound to. This way the exporter only sees the simple rig.
We have a character with a custom rig (jaw, eyeballs, eyelids, eyebrows) on top of a standard Mixamo rig. Two problems: 1. When playing a stock Mixamo anim, the character's jaw now falls slack. 2. Eyebrows and eyelids can be animated in Maya but the animations doesn't come through to Unity. We use Maya 2012 and Unity…