You have several proble ms.
The first thing:
i would do is disconnect the Batteries,
Second: i would connect the new converter directly to the batteries after all it is battery charger, 3rd Thing :
I would connect a power cord to the converter, and plug into an extension cord run from the garage or house, now you are charging the batteries directly,
Keep charging until voltage at the batteries is 14.2 volts
Then unplug the converter from 120vac, leave connected to batteries , take periodic voltage checks at the batteries,
This will let you verify the batteries good or bad
It sounds like you have multiple problems, your demensional inverter has a pass thru function for 120v, the contacts might be stuck, and not switching to inverter output, or it could just be worn out, its also possible you have a shorted cell in a battery so they do not get fully charged , and then loose what charge was obtained when charging is stopped , you haven't run. The engine long enough to completely charge the batteries, and verify whats going on.
I can explain it to you.
But I Can Not understand it for you !
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