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Hi, I'm Robert. Welcome to JustAnswer. I'm reviewing your question now and will post back with your reply momentarily.
When you are plugged into power the power converter or inverter will charge the batteries. IF the batteries were in poor condition they might not be able to accept a charge. So the first step is to have the batteries load tested, most auto parts stores can load test them for free. IF they are bad they will need to be replaced. Make sure all your cables are clean and tight as well, no corrison on the cables.
When you are plugged in you should see close to 13 volts at the battery if they are in good condition, if not then you may have a problem with the power converter or inverter. An inverter will allow you to have AC power when you are unplugged, a power converter will not do that but they both will charge the batteries. Sometimes you might find a breaker between the converter and battery, below is a picture, as a general rule they like to stick that 18" from the battery and it can be on the frame. IF you find any check for voltage on eah post to chassis ground, if one side reads much higher then it is the problem.
I would really start off by having the house batteries load checked, just mark the cables and or take a picture of them before you remove them.
Hope this helps if you have more questions or concerns please feel free to contact me.
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HI was just checking in to see if my answer helped??? IF you have more questions or concerns please feel free to contact me...