So I pulled my head out of my butt and disconnected the DC supply wires from the servo amp's and the ground fault lights are off with the other two plugs still connected.
Like I said, the caps have been replaced, and have been verified to be good (as I was informed being brand new doesn't guarantee anything). What else could've failed when the cap went bad? The two shunt resistors measure lower than their rated 3000 ohms, but they're identical resistance, and one of them looks to have gotten hot (which is the one which was across the failed cap).
Is it possible that the bridge rectifier is shot, and that's what's causing the error light? What else in that end of the circuit could've been damaged to cause this issue?