And yes, the separate power feed is used to power the relay and the lights.
The caravan side of things is a little more complected and I think you need a 'split charge relay' This is used to charge the caravan battery and run the fridge in the caravan.
https://www.amazon.co.uk/MAYPOLE-UNI...e+relay+towing
Or both in a kit
https://www.amazon.co.uk/Double-7-pi...e+relay+towing
I'll be honest, I never bothered with this when we used to own a caravan, I just ensured the fridge was turned on and cooled down (from the mains) the night before and everything stayed cool until we got to the site.