hi, sorry can't help with the wiring as mine is 2007 and 7 pin x2, but am intrigued that you are happy to tow with the fridge running on gas.
whilst not illegal it's nevertheless fraught with danger so assume you're on board with the possible risks of a naked flame !
btw are you aware that you MUST turn it off if pulling into a petrol station ? and not when arriving at pump but before you enter.
also if the fridge is of the absorption type it's very intolerant of not being level and being subject to going round corners, so your food may
not be as cool as you hope on arrival ! it's easier to precool overnight on 240v before travelling and pack food in a insulated box (with ice blocks)
carried in the car (less towed weight !!)
Jmo and i hope you never have a shunt on route...🙂