Your fridge on DC uses 15 or so amps?
Your alternator puts out 160 amps. Mine does.
Lots of extra power. Powering through the TC plug and 12 gauge truck wires will introduce a lot of I/R drop and loss of power to the fridge due to the small wire size.
In the Truck Camper University section, there are several good threads on how to install a 4 gauge wire from the truck alternator to the TC batteries and electrical making a marked improvement. It did for me. Enough so, I don't need a generator, I just run the truck for a while to charge the AGM batteries if the solar isn't keeping up.