I don't know if I'd attempt to drill it while installed. I would be afraid of damaging the pilot circuit. Anything you do to push down on the head is risking jamming the taper into the seat.
Do you still have the stock PMS screw installed (the kind flush with the surface)? You may be able to carefully epoxy (or possibly JB Weld) the head and back it out gently. I'm not sure how you would repair the threads, but I would replace the PMS with Maxair MaxMix style screws.
Take a look at this Wiki to see what you're dealing with here: