I think you'll have a hard time finding anything like this available, certainly not as a packaged deal. I'm not aware of any RV charters that include a driver, or at least not for anything less than utter opulence. You would presumably have to hire a driver and rent the RV separately (or hire a driver who is willing to rent the RV). I am not sure too many drivers would be willing to accept this sort of job due to liability and possibly legality concerns.
Practically all campgrounds are run with the assumption that you are staying nights, not days. Arrival hours are in the afternoon/evening and departure hours are in the morning. You may very well have to pay for an extra night at each campground.
My hunch is that you wouldn't sleep particularly well in the RV while it's in motion. It would be OK for a night or two, but after a couple of weeks, you'd be beat. I may be all wrong. The younger kids might do better than the rest, actually; many kids seem to be able to sleep decently in impossible positions and situations.
My free advice, and worth every penny, is to plan to see fewer things in a more traditional way. A minivan rental and motel/hotel rooms would be considerably less expensive and less fiddly and probably more restful.
