IMHO, I would do a rewire as opposed to running a 'ground' (EGC) conductor.
Issues with conformance with the interpertations of compliant NEC methodology may arise with your plan. One may be that the EGC be run within the cable/raceway with the circuit conductors. Another may be the method used to enter the boxes with the single conductor. A conversation with your AHJ may be a good point, before you proceed.
As to the insulation, you should check with the mfg to determine if it is not detremental to the outer jacket.
Again, IMHO, I would sell a rewire. The sheetrock is down!
