Does your battery recharge while on shore power? If so the TT has a built in charger. Just connect your trailer to the generator and your battery will recharge.
It may take a while depending on how big you battery bank is and how far down they are. Also while charging the load on the genny could be 300-500 watts so you may not be able to run appliances such as the microwave.
I have a Honda eu2000i and an external 30A charger for my AGM battery. When the charger is running full tilt if my microwave starts it will trip the overload on the Honda. For me it's no big deal, I just yank the charger plug til we're done with the microwave. On a built in charger it would be a pain.
