Refrigerator operation

Just et it to Auto and leave it.
This way it will switch from AC to Propane itself.
As for inverter if realy hooked up that way. The inverter would have to be turned on. A separate switch.

If this is a two way absorption refrigerator (likely unless quite old), 12 VDC is ONLY for running the PC board, gas valve and ignitor.

It can use 120 VAC from shore power or generator for the "heat" source. As to whether it can also use inverter power depends on how the outlet the refrigerator is plugged into is wired. Easy to tell on some models-- look at the control panel and see if it still shows to be on 120 VAC or has defaulted to LP. Another way to tell is to disconnect from shore power and generator off. Go to the outside refrigerator access door. Unplug the refrigerator 120 VAC plug. Plug in anything such as a light, hair dryer, etc that will tell you that there is power to that outlet.

As far as what heat source to use going down the road, that is open to debate. The most cost effective way to produce heat is with the small LP flame. Much more efficient than turning gas or diesel into power in the engine to power the alternator to produce electricity to power the 120 VAC heating element. But, there is a small risk with any fuel source on. Most find it an acceptable risk-- your choice for you IF the inverter will power the refrigerator outlet.