It’s common to mount a battery tray on the tongue.
If you mount it inside, I recommend using a battery box and venting it with one vent through the floor, and one through the wall to expel gases while charging.
In my truck camper, I use a 12v float charger connected to the 120v shore power so that it tops off the battery when plugged in.
A 120v relay disconnects the battery from the “house” power while at the same time connecting the converter. This prevents the converter from cooking your battery and isolates the two systems.
The relay is less than $20 and float chargers are fairly inexpensive.
It’s a DIY way to add onto what you already have.