You have some bad wiring somewhere, has it been alright before? if yes it could be the car but may be the van.
Sorry a bit ambiguous but that the way it is, if the van is pre-used the previous owner may have rewired to suit his badly wired car socket, however as you've just had a new tow bar fitted and we must assume wiring it could be your wiring at fault
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Could be a loose wire in the plug. Have you tried disconnecting the 12S as that's the only plug with a constant power supply, which is also on the same pin as the fog light which makes me wonder if they have been crossed over. Does the fog light stay on with the car unplugged? if so, must be the caravan at fault.
We have just had the same bother, had a towbar fitted at home by a proper firm but the fog light stayed on all the time while towing. They came back out and found that the fitter had wirred a perm live to the fog by mistake, so could be wirred wrong or a loose wire touching.