Really what I'm looking for is an allen key-type tool that won't degrade
under stress like a full allen key will and/or one that I can apply either a
shock to (impact or something) or lots of torque to remove the flat head.

If it's a 1/4" hex, just get a 3/8 square drive allen bit to fit a
standard ratchet. Cut the hex tip off to just a little longer than the
recess is deep. You will break the screw before the bit twists.
If you can't find the bit ready-made, get a socket and cut off an
allen key, glue it into the socket.