Inverter/charger combos - way to expensive and require hardwiring into the camper's electrical system and the removal of your current charger/converter.
On the flip side, you also need to understand there are also 2 kinds of inverters -- true sinewave and modified sinewave, true being more expensive than modified.
Without getting too technical, the difference is--
True sinewave - as close as you can get to true household 110v. Puts out a nice clean 60hz (cycle) sinewave, but cost more.
Modified - Puts out a harsh 60hz square-wave. Much cheaper, but not good for sensitive electronics, and especially medical equipment.
I would look into Xantrex True Sinewave inverters. They're about as clean as you can get for a reasonable price.
Keep in mind, depending on what you're running off them, inverters draw a LOT of amperage from your batteries. I run 6 golf cart batteries, and that gets me easily 3-4 days of running anything I want before I need to consider recharging. But again, that's SIX batteries.
'15 F450, 30k Superhitch, 48" Supertruss, 19.5's, Torklift Fast Guns
'12 Eagle Cap 1160, 800watts solar, Tristar MPPT, Magnum Hybrid 3k Inverter
'15 Wells Cargo 24' Race Trailer, 600 watts Solar, TriStar MPPT, Xantrex 2kw inverter
'17 Can Am X3 XDS Turbo