ResearchDirector: If you teleported, would it take the C4 with it? And the bag?

Adhesive properties of the explosive under conditions of time-space tele-distortion are unknown. No guarantees, and "no guarantees" is bad for an untested hypothesis. You have to go with the assumption that at least the explosive would come along with the mecha.

ResearchDirector: Drill your own cutter off, and then teleport.