I think you need to look for something else that is ( or may be) charging the batteries. I too have a Xantrek inverter used to power my TV, and no it does not charge the batteries. You need to look for your power CONVERTER which would be used to convert 110 power to 12v for charging purposes and to thus to feed the 12v system while plugged in. It will most likely be a smallish box with perhaps PD92xx for Progressive Dynamics on the faceplate.
The inverter does not need to be off while connected to shore power, but it is not required to be on either. In my case there is an additional box by the inverter that switches between the 110 source ( if present ) or output from inverter. You may have one of these also.
You will want to turn the inverter OFF while in storage or not being used, as well as perhaps the house battery disconnect switch (assume you have one) to reduce the chance that they will drain. Some RV’s will NOT charge the house batteries even if plugged in if the disconnect switch is active.
The inverter would not be charging the batteries, that would be the job of the converter. Your owners manual should indicate if your house batteries are being charged while plugged in. Some do, some don’t.
I have no clue on what solar equipment or size you would need.
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Tom
2016 Newmar Bay Star Sport 3004
2021 Jeep Gladiator Sport Willys
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