Well, drilling out the non-existent rivets didn't go so well , but I did futz with it some more to refresh my memory.
I discovered the island lock solenoid's locking pin had come unscrewed partway from the plunger, causing the pin to not retract as far as it should. I thought, that had to be it! But once I tightened it back up, it started resetting more often than ever!
Turns out the switch was now reliably not deactivating when the solenoid was not energized, telling the game that the pin was down when it in fact was not. So guess what happens when the island is locked but the game thinks it isn't? Yup, it tries to turn the island to seek the next locking position, but it can't turn, so the motor overloads and we get a reset!
A simple tweak of the switch blade and it hasn't reset since! Booyah!