Broke the head off a bottle cage bolt and can't extract it
I was going to put some water bottle cages on an old Trek Hilo 1000 that I sometimes ride in the morning when I have time. 3 of the 4 screws came out with no problem, the last one fought until I over-torqued it and broke the head off (something I never remember not to do even after dozens of times).
Now, the screw is stuck in the frame, and I am exhausting all the ways I know how to get it out. Because of the design of the frame, the hold is recessed enough to where I couldn't use pliers of any type to grab anything. It also made it impossible to use the hacksaw trick. I have drilled out the center and used screw extractors with no luck. It looks to me that the screws are all anodized aluminum, and the metal is so soft that the extractors just take away more material.
Without being able to head to a Trek dealer until the end of the week, it seems there is no way to get the screw out without destroying the hole. The only other thing that looks possible is by removing the inset piece that I have attached a picture of. It moves slightly but is probably epoxied in and/or held in place by the paint on the bike itself. Anyone have any other ideas that might work?