Success! After a broken punch, a broken drill bit, and a few trips to the hardware store for ever-smaller tap wrenches, I got it out.
Originally Posted by
rydabent
Combine several suggestions. Use the lightest oil you can find, even cut it with kerosene and let it soak. Tap in slightly, and warm the tube with a heat gun or hair dryer until warm to the touch, but not ruin the paint. Then if that fails center punch and use a left hand drill or an easy out. Above all be patient.
I used a version of this. Had to step up from a 5/64" bit to a 3/32" bit to get a good center hole and then finished with the 5/64" bit. Dumped a bunch of GT-85 into it before trying the extractor. Re-tapped the boss just to be sure that it was still functional (the bit had jumped around a bit when I broke it). Thanks for all the suggestions/tips/tricks and encouragement! All three bottle cage mounts accounted for!