I'll try to take the switch off, give it a shake and reset it then. Won't be able to test it until we get a new battery fitted, but maybe we'll ask if the garage can fit a new battery, then test it again before replacing anything else. Thanks for your help

Change of plan - we've arranged for the car to be taken to the garage on Monday, they'll check (and charge, if needed) the battery, check the wiring in the back and replace the pump & sender. I forgot to mention that our fuel gauge doesn't work and hasn't worked since we got the car, so that needed doing anyway.
Thanks for all your help - hopefully the issues will be resolved when we get our car back, and if so I'll edit this with the solution in case anyone else has it in the future.