If shore powers the battery charger, and that's not clear because some inverters are combination inverter/converter (charger), others are just inverters, then leave the Aux. battery switch ON so the house batteries charge. If unplugged or not on generator power, turn that switch OFF unless, as mentioned, you need DC power inside the RV.
My advice for the inverter is to turn it OFF if not using it to produce AC power for your RV. The inverter draws power even if not inverting if you leave it ON without shore power connected. Connected to shore, you don't need it unless it is also a charger. Disconnected, you only need it for limited use of AC power. In storage, turn it OFF.
From you picture, your inverter does not appear to also be a charger because there is no LED to indicate charging or in charge mode. You may have a separate charger, however, and that will need to have the switch ON for it to charge the batteries, in most cases.
If you can find the inverter and post the brand/model, we can look up the manual and clarify its functions.
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Larry, Eileen, and Finley
2004 Alpine 36FDDS
Third motor home, first Alpine, no need for another.
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