Can I plug in a regular battery charger into the inverter and charge the batteries while using the inverter to keep everything working together inverter suppling power to the battery charger and the charger charging the batteries?

Yes, it will work, but not for long.
There are losses in conversion from AC to DC, Through transformers, Inverters.
Sounds like you are after Perpetual Power, not yet possible, I don't think, That would be like free power.