Melissa, if I was determined to use a gear puller, I'd try to find another screw that fits well into the shaft, and preferably a square of Phillips-head one so that the puller is centred in the screwdriver recess, rather than use the original screw and bugger it up. The original is likely to be a brass screw, therefore not very strong, and the shaft is likely to be brass too. Threads in brass are EASILY stripped.

I'd consider getting someone to grind/file off the first threads (on the new screw) leaving a 'tit' on the end in hopes that you can bottom-out the 'tit' and NOT work on the thread. Careful comparison of the thread depth versus hole depth needed here.
I'd also use a hose clip around the legs of the puller so that it worked on the boss of the handle.

As several people have said, use heat, maybe from a borrowed gas blowlamp, ONLY (as far as possible) on the handle. Brass expands well with heat, and it should expand away from the shaft.
Hope this helps.