The feed to the fridge is never permanently live for obvious reasons. It uses pin 10 which will only supply current with the engine running .
When we crossed to France or Spain with previous Jeeps I always disconnected the electrics as did not want a flat battery on arrival at the port. I came across this article regarding the Sterling Wildside.
The Sterling BBC1225 'Wildside' battery-to-battery charger provides fast charging of your caravan or trailer battery via your vehicle's towing socket. It takes the low voltage at the towing socket and boosts it to provide the correct, multi-stage charging profile for your battery. This unit is particularly suited to those who prefer touring, and so don't have frequent access to campsites with mains hook-up, and/or those with vehicles whose engines are Euro 5/6 spec.
The traditional method of charging directly from the alternator via the towing socket offers very poor performance because of the small cable and long run lengths used. This results in large voltage drops meaning zero to little charging of the auxiliary battery takes place and often the on-board fridge simply can't operate at the low voltage provided. Electrical demands have increased in modern caravans (caravan movers etc.) meaning that good charging has become even more important, however, with the introduction of Euro 5/6 engine vehicles with 'smart' alternators, the charging performance has, in fact, become even worse. This is because these alternators reduce their output voltage to save fuel when the starter battery is nearly full, which amplifies the already poor situation with voltage drop and effectively renders the charging system completely redundant.