The reverse light is powered from the 12s (Supplementary) socket.
Behind the lh rear lamp cover there is a spare connector with, I think 3 wires.
One of the wires is a reversing light feed, another is a ground-but low amperage capacity and the third is an "Ignition on, live feed. The Ignition live and ground are used to power the towing relay for powering the Caravan battery charge, Fridge supply and 12v "Ignition off" for caravan interior lights, pump etc.
You have correctly run a fused 12v feeder from the battery to the rear end which should be suitable for carrying around 25A and allow for voltage drop in view of the length of run. 4mm or 6mm sq. should be ok. The relay can be mounted behind the lh access panel for the rear light cluster. A 20 amp relay with double N/O & N/C contacts which are available from RS components or similar supplier so the fridge and battery charge can have their own contacts fed from the single battery feed. The Ground wire from the 12S socket should be terminated by a screw attachment directly to the car body.:behindsofa: