Visitted the shop where I had bodywork done last time and they were kind enough to borrow me a fully charged battery and a set of spanners so I could do the swap. As soon as the new, full battery was in, the fan fired right up and I fired the engine up and drove the car off to the shop.
I tried swapping both thermostat relays to no avail (on borrowed parts. Some people owe me favors), so it can only be the AC relay, which has already done this in the past. Still, it's not an extensive repair, the car should be ready by tomorrow afternoon or the day after.