So I am not sure if the GFI was on the fritz or not. The wiring in the box (well, at least the grounds) were a little suspect, so it was nice to clean things up in there anyway. And we are making a new car for the next competition anyway so no need to worry about putting the GFI back in to this particular circuit.
The problem turns out to have been one of the emergency stop buttons. There is one located behind the brake pedal such that if the brakes fail the pedal hits the switch and shuts the car down. Our problem has been a long brake pedal (or a switch not placed in the correct place if you prefer) so the switch was tripping during hard braking.
The team swapped the leads on the switch such that instead of the "go" position being fully rearwards, closest to the brake pedal, the "go" position is now in the "stop" position, as if the brake pedal had already hit it.
They neglected to give me this helpful bit of information and I have been trying all this time to get power with the brake switch in the logical "on" position as if it were functioning as a brake overtravel switch.
That is what I get for assuming the logical conclusion...
