Your needs indicate a class A coach. I know you are familiar with the tourist busses you see all over Italy, that is what you will be driving. That means no dirt roads and staying in private RV parks that have sites large enough for it to fit in. That has advantages though because you will have water, electric and sewer hookup which for 6 people you will appreciate that. Also have cable television and other amenities like a play ground for the children.
You are going into LAX so the closest park you are interested in is the Grand Canyon. Nice RV parks to accomodate you and civilized enough to provide bike rentals and rafting adventures and hiking, everything you are looking for. Google maps shows it to be about 7 hours away from LAX, not a bad drive.
