Only one of the contacts in a 7 way is 12v, and often just enough of a gauge to run the trailer's lights.
Can you do it? Sure, but it might not work as well as you'd like and seems like an odd workaround to just running a dedicated line from your battery. Then you'd be stuck with a 7way on your fridge and not be able to plug it in to any other 12v outlet.
Most 12v fridges have a low power setting, that when it thinks the battery is getting too low in voltage, will slow down the cooling of the fridge so you don't kill the starting battery. I have no doubt that the 7way gauge on the 12v wire may be insufficient enough that it will fool that low voltage sensor. It's a common problem on some Jeep's rear 12v outlet.
Australians often run a completely separate, and heavier gauge, 12v line from their battery to the hitch, ending with an Anderson plug, to which a similar line on their trailer can plug into, avoiding using the 12v line in the 7way at all because it is not enough to power the add'l 12v needs on their adventure trailers. I'm getting ready to do the same with my van and trailer.
The best thing to do, really, is get a $42
ARB fridge wiring kit, which has 20' of 10AWG, an inline 15amp fuse, eyelets ready to connect to your battery, and a nice surface mount 12v outlet for the rear of your vehicle. And, if you change the 12v plug end on your fridge to an
ARB 12v plug, "the cigarette plug tip can be removed and the special two pin plug can be screwed into the matching ARB power socket for a secure connection."
Works great at keeping the plug from vibrating out on washboard roads and rugged trails.