On leisure batteries, and fridges

Hi

Have a Bailey Pursuit 430-4, but my car only has 7-pin electrics. Was going go get a split-charge relay fitted and a secondary socket as I'd been told that despite their being a leisure battery, the fridge wouldn't keep cool running on that while towing.

Which seems to defeat the purpose of having a battery a little. Can anyone definitively tell me whether the fridge will tick over on the battery until we get hooked-up on site please? The auto-electrician I spoke to about the wiring reckoned it'd be a first if the fridge didn't work off the battery.

The owners manual is a little vague on the situation and likewise Bailey weren't overly clear when I asked either, unfortunately.