John, Sounds like your batteries are not getting charged properly but we'll need more info. In the meantime...if you have a VOLTMETER With your genny on...put the voltmeter on the terminals of each battery and take a reading. You should see something ABOVE 14V if your charger is working. If you get something in the 12V range or less...then your charging system is not working. It could be cables OR the charging section of your inverter or converter.
If you ARE seeing over 14V then my guess would be that you are not nearly charging your batteries with enough amps or long enough to fully charge them to 100% which MUST be done.
Suggest you also try this same experiment with the genny off and the engine on to see if you are getting a charge from your alternator.
To be of further help, it would be good to know what make model inverter/charge you have..what make model batteries you just installed as well. I assume both are coacch batteries rather than chassis/start battery.
Cam... Georgetown 280DS 2015
Vespa Lx150 2013 Stella Auto125 2014