My truck has a circuit breaker at 25amps for charging the trailer batteries while traveling. That is 300watts of charging power theoretical, however there is some loss in the wiring. What does the residential fridge require?
The refrigerator is powered by a 1000 watt inverter that draws from the batteries. At startup, compressor demand may briefly exceed 1000 watts and then it probably draws 6 amps or so, maybe around 720 watts while the compressor runs. If it ran continuously, even if you got 300 watts from the tow vehicle, it would not be enough to keep the batteries charged. But the compressor shouldn't run continuously. So let's say it runs 15 minutes per hour. That would be 180 watts. Add some for the inefficiency of the inverter and for startup surge and we'll call it around 225 watts average. If your truck and trailer wiring gets you 225 watts to the batteries, you should be able to keep up while towing. Start towing with a full battery charge and end up with a full battery charge. But then you've got 16 hours not towing and the refrigerator still needs power. Since it's usually cooler at night, the compressor probably runs less. But it still may need quite a bit of power from from the batteries. And if you're unhooked from the TV, it's not getting help from the TV batteries. That's why you might be ok for a one night stay at Walmart, but probably not two nights.
Btw, this is why it's a bad idea to leave the umbilical connected to the truck overnight. If power goes through the connection with ignition off, you could wake up with dead truck batteries.
Please note that these aren't exact numbers. Refrigerators vary. Battery conditions vary. Trucks vary. Temperature varies. This is just intended to get into the ballpark to illustrate how it all works.