Do you have a Dremel tool handy? I have salvaged some broken scope mount screws on some pretty expensive rifles by using a Dremel and a tiny diamond-coated abrasive bit to core out the screw without damaging the threads, then using a tap to chase the threads. Some of the diamond dust bits are as small as 1/16", and with a set of magnifying goggles, it's possible to be very precise with the core-drilling of the broken screw, particularly when the pilot hole is a little off center. Check Ebay for the diamond Dremel bits. They're pretty inexpensive.
The drum is probably cast iron, so don't worry about annealing it.
Jerry