Depends on a number of things.
Most tow vehicles only provide a trickle charge IF the vehicle has a 7 pin connector and the correct fuse is in place.
If you have the 3 way fridge set on 12v, this could be a problem.
My bet is that you don't have your vehicle setup right.
Might help if you posted what it is.
and Zoe the Wonder Dog(R.I.P.)
'07 Roo 23SS pushing an '07 Chevy Avalanche
Equalizer WDH and Prodigy BC