Eric -
Please let us know how the Grabits work for you. Unfortunately, I think Keith will turn out to be right. In order to make the conical hole, the Grabit expects that a hole of some kind already exists, though it may be a very messed up hole (stripped Phillips head, for example). The burnishing end of the Grabit simply improves the shape of that hole, so that the extractor end can grip the sides of the hole. Take a look at this link to get a better idea of how they work.
http://www.4grabit.com/Default.asp?bhcp=1
On the other hand, drilling a hole in the screw shank shouldn't be impossible. It would be easy to do if you could clamp the whole TM on a drill press table, but obviously you can't do that. However, a trick that I have used with moderate success is this. Get a piece of acrylic or plexiglass, maybe 3" square. Put a small bit in your drill and drill a hole in the middle of the plexiglass. Now unchuck the bit, and re-seat it DEEPLY into the chuck. The amount protruding from the chuck should be equal to the thickness of the plexi plus about 1/8". Now hold the plexi FIRMLY against the side of the TM, so that the predrilled hole is over the center of the screw shank. Put the drill bit through the hole, and slowly drill into the shank. Not much forward pressure on the drill. The plexi keeps the drill bit from wandering, and the fact that only a small amount of the bit is outside the chuck keeps the bit from bending. This jig would be better if made of metal rather than plexi, but you need to be able to see through it, to put the tip of the drill bit in the right spot.
This is a practiced art, and the fact that the plexi will be resting on an uneven base - the piece of trim - will complicate it. It may, in fact, turn out to be a two-person job. But with some care, and maybe shimming up the plexi a little bit so its surface is better supported, it may work. Once you have an initial hole in the screw shank, it is a lot easier to make the hole deeper and wider, and at some point the Grabit should be able to (ahem) grab it.
Bill