Okay, problem solved.
For future reference, here's the relevant information:
The screws have a 5-40 pitch thread. Best solution is to get a hold of a typical 5-40 hex socket cap screw w/ 3/32" drive, and longer than it needs to be. It's better to be a little longer on the upper screw, because the one that comes with the gun has maybe three threads holding it in. The lower screw needs to be exact, because it sticks out of the other side of the frame.
Anyway, you take a 5-40 hex socket cap screw and reduce in a lathe the diameter of the head from 0.200-0.205" to no more than 0.162". Then you shave the bottom of the socket head to thread diameter, to adjust the height of the screw head, and so the drive doesn't get too shallow.
Then cut the screw to the appropriate length (a little longer on the upper screw if it's like mine, at 0.312".) The guy that did it for me also shaved a little of the top of the head to get it close to flush, but we decided not to go any more, or I'd be in the same boat when I went to turn it again.
Also, in case this part isn't obvious: The socket cap screw can't be nylon, brass or zinc. You want a good steel part.
So there you go, best solution, and you get a larger socket drive. Now all I have to do is blue it.