Do you have a drill-press ? I've turned a few items on my drill press with satisfactory results.
Other than that, I would place the cap (open-end pointing downward) over a small-diameter woodscrew pointing straight-up, and balance it as best you can. Once you find that spot, press the cap downward on the nail to make a dimple in the plastic cap. Then, choose drillbit slightly smaller than the woodscrew and drill your first hole at that spot.
Screw the woodscrew into the hole in the cap, and try spinning it like a top. If it wobbles too much, you might be able to nudge the hole in a different direction as you change to larger drillbits getting to the final hole-size.