GFCI current sensors are apparently easily triggered by radio waves.
In an apartment I used to live in, the bathroom GFCI was apparently wired backwards. When wired backwards, it can still detect leakage current since that is a common mode AC current, so direction is not relevant. The problem with backward wiring is that opening the circuit does not disconnect power to the solenoid that opens the circuit. If for some reason the triggering current remains, a backwards wired device will keep operating the solenoid.
I found that 1 watt of RF in the amateur 2m band on a 1/4 wave or 5/8 wave antenna within 3 meters of the wiring leading to the backward GFCI would cause it to trigger, and it would keep buzzing loudly for as long as the RF was present.
RF can trigger these things. When the power wiring is acting as an antenna for nearby RF, it will be in common mode (both conductors conducting in the same direction at the same time), which will appear like leakage current. A simple rectification circuit to detect a voltage from the current transformer will operate over a large RF frequency range (e.g. from a few Hz to a GHz or more).
The arcing from the doorbell switch could be a long enough RF pulse to do this. Or the bell itself may be doing it.
RF filters that block common mode currents in the RF range (but not in power frequency range) could be used to stop such issues. Just be sure the filter used does not block 50-60 Hz common mode, or it could defeat the proper ground fault detection.
