Are you sure it's circuit related?, my thinking (but without seeing the code) is it's interrupt related and you are being swamped with interrupt requests due to the high level of rainfall resulting in the watchdog being triggered on the ESP8266.
If you don't get the crash with low / no rain, it's unlikely to be the circuit. Your increase in resister for the pull up will only come in to play if the distance between the sensor and controller is very long and your wire is quite thin and that would result in missed activations not increased / crashes. You can test this by disconnecting the RG11 unit and manually joining the wires together to simulate a pulse from the RG11 relay.
If you disconnect the RG11, does the ESP still crash? What does it output on the serial port?
I have a RG15 and it can be sensitive to the input voltage causing brownouts, but that would be unlikely to lock up your ESP as it's a separate unit but should't be ruled out just yet though 2A should be more than enough for both devices to co-exist.. (RG11 is 55mA max @12V input with everything on from the datasheet)