No way to know the answer.
need more info...
Inverters are usually turned on manually. Keeping them on all the time still uses electricity even if nothing is using the inverter for power, inverter draws power just being on, so keep it off until needed.
The way to know if it comes on automatically is to do a test. If it comes on then it is auto, if it needs to be turned on then it is manual. It is just that simple.
The following is assuuming yo have some soleniod switches, not a box that does it all electronically:
Automatic switches are typical for letting the "camper" choose to use generator power or to use household/shore power. Typically the default setting is for the electricity (120 household current, not 12v DC) to choose the generator setting, but when plugged into power the "switch" will choose shoreline power. Only one can be chosen and the reason is so that the generator does not get fed voltage when the camper is plugged into shoreline power (which would burn it up).
There is no reason an auto switch cannot be used to select inverter over another source of power. Any two sources of 120 household current can be wired into the selector switch and the switch will "prefer" one over the other depending which is connected to the "preferred" location.
To do 3 power sources you would have a switch that selects shoreline or generator, then you would have a switch that selects generator or inverter. The first switch would prefer shoreline power (or maybe prefer generator), and the 2nd switch would prefer generator (or shoreline). So if no shoreline power but there was generator power then it would switch to the generator side, but if the generator goes off (and the inverter is on) that 2nd switch would switch to the inverter.
If inverter is turned off then the switch would go back to the generator setting even if gen is off, and the 1st switch would go back to shoreline even if there is no shoreline power - those are the default settings.
Also, the inverter would need to be isolated from the AC unit and the refrigerator 120v circuit. Possible to keep the fridge on 120, but most likely you would want to save the batteries and use propane for the fridge as the fridge would kill batts pretty quick. Better to use 12v DC if you have to run the fridge off batteries, but that is just slightly better, it will still kill batteries... So the AC circuit will need to be on the shoreline/generator side of the switch in order to isolate it from the inverter. I'd have to ponder how to wire this for auto switches...
I had a camper set up to use all 3 voltage sources, but it was all manual. The circuit breaker was paired up to only have generator on while shore was off or it would turn gen off if shore was on. The inverter was on a switch that the outlets went through, but not the AC or fridge...
Unlikely you have such an elaborate system, but keep that in mind when looking your system over.
