I've been happy with my 4 gauge alternator to TC batteries connection. 2000 f250. If I were to power my fridge with 120V from the inverter, to keep things cheap and simple, I would merely take the shore power cord and route it to the inverter. Be sure to switch off the AC breaker and unplug your battery charger. The perpetual charging scheme has already been disproved. My inverter is only a 600 watt pure sine. A poor man's manual transfer switch if you will. My alternator is 160 amp output. The TC 4 gauge is fused with a 30 amp auto fuse which is probably costing me performance in current flow. The fuse has never blown and usually supplies 14.2 volts which drops to 13.4 after a while. I also have about 240 watts solar feeding the batteries in the TC via 6 gauge from the controller.
I'll have to actually try it sometime since I'm completely happy with running the fridge on propane.