Running new electrical with drywall removed, but insulation installed
I find batt insulation easy to pull out and re-install. Drilling near existing fiberglass (guessing that is what you have) can lead to strands getting caught in the bit and spinning.

I recommend pull out the batts of insulation, do the electrical work and they reinstall the same batts you pulled.

Also assuming you have vapor barrier or retarder on the face (probably what the staples are in), you can try saving the vapor barrier but it may rip. Just add another layer of Kraft paper or 6 mil poly sheeting to replace the old stuff.