use generator/inverter to charge batteries and power trailer

Let's start with a converter will charge your batteries from 120 volts and an inverter will use amps from your battery to give you 120 volts.

If you plug your RV into your generator to use your hair dryer your converter will be charging your batteries at the same time. There is no need to use a separate battery charger. It will take 3-4 hours to charge the batteries depending on how much you discharged them the night before.