Dual 12volt RV/Marine batteries should easily make it through one night unless you are using a lot of non-LED lights and are near freezing temps outside.
You probably need to equalize and then fully charge to the batteries. Your stock converter might not be up to the job of doing the recharge if it doesn't take your batteries to 14.4volts for a couple hours during that recharge. You should recharging for at least 3 hours at that 14.4 value everyday. If your converter only gets to 13.6 then you would need to run the generator for much longer, like 10 hours to get to 90% charged.