The least expensive way to handle this while you're traveling is to use a set of jumper cables (the cheap ones will work here...maybe pick up a set at a yard sale or thrift shop). Whenever you park and plug into shore power, just jumper the battery sets together so both sets, chassis and house, get charged by the converter.
Meanwhile, find a float charger 2 Amp or larger (WM has them at $20 or $2 at a thrift shop) and connect that to the chassis battery, plug it into an extension cord or nearby patio outlet.
Just remember to unplug things before traveling, and that the 2 6V batteries are wired in series and you want to connect the jumper cable to get 12V.