IF your getting into refer outside-you don't need to reroute inside switch-just add another switch on the 12v input to turn off. Turn back on when ready to start truck (or after starting). You have a minute or 2 before the check light comes on if AC not available yet.
I'd do the relay (see previous post edit) but on the 12v side to turn on/off refer (control from cab, keeping fridge from looking for alternate source). If you want to make it auto could power the cab 12v switch that triggers relay to an ignition source. Truck turned off, power to relay cut/refer off. Start truck power to relay/fridge comes on. I don't know what controls your inverter but if it comes on before fridge goes to check-done.
Only problem with cab controlled relay- need to add a bypass when camped.
