If all you are running is LED lights
, a standard group 24 RV/Marine battery
should last forever.
An inverter converts 12 VDC to 120 VAC. If your interior lights
are currently 120 VAC, I suggest that you just change them all out to 12 VDC LED's and not fool with having to add an inverter as it will just put additional load on your battery.
That said, you usually can't add a new battery to an existing Old battery, it's recommended that you start out with two new new batteries to obtain proper charging on both.
But again, if it's just lights, one battery and LED's should do the trick.
The next question is about how the battery is currently being charged, but that's another topic all together.