Old school way of doing it would be not installing another battery, and just unplugging it at night.
These Edgestar Fridges have reports of being very sensitive to voltage drop. Either not working, or just working poorly.
12 awg sounds fine, but use 10awg and make sure the spade terminals on the compressor controller grasp tightly. Ciggy plugs, even the good ones, are not that great of a connection. Most factory installed receptacles use 18 awg, and long lengths of it, and need to be bypassed for powering a compressor fridge
I'd have a non voltage sensing solenoid to isolate a second battery, and no way would I unplug the fridge overnight. I'd have a 10 awg wire with an Anderson connector with as short of a run as possible to the battery, fused at the battery. I'd make another connector with a ciggy plug to plug into the Anderson connector for operating the fridge in a different vehicle.
Even connected to the alternator, the second battery will still need to be fully charged by a different source. Squeezing in that last 20% takes forever, and is required weekly for battery longevity. So have a way to plug into the grid and charge overnight, or resign yourself to replacing the aux battery more often, after it lets you down at the worst possible time, because Murphy was an optimist.