Just for clarity, keep your terms straight. I think we know what youre talking about, but others.....
Converter converts the AC into DC and should keep a charge on the batteries. An inverter, takes DC power and spits out AC.
Other than that, I couldnt say much without knowing some numbers. I think my converter had a high 13 on the battery, 13.8 or something. When the engine is running and the alternator is doing its job, it went to about 14.8 or so.
I would also recommend that you put a current meter on the battery with everything off (should be zero), then with shore power (should see a charge current). depending on the state of the battery, this could be large or small, so use a meter that can handle about 10 amps or you may blow the fuse in a smaller meter.
FYI, our GS has a pretty good (sounding) converter, it should tend the battery quite nicely, its not just 12V on the battery. If the battery is in good shape, you may only see a small trickle charge, which might be intermittent and hard to pick up.
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