Thanks to all who responded.
I ended up purchasing a Swagman Escapee bike rack. It is motorhome approved and it is a tray type. The bikes mount very securely. It will accommodate 2 bikes up to 60 lbs. each, which will be great if we later upgrade to e-bikes.
I bought it from etrailer.com. It arrived with no damage and was easy to assemble. It is a heavy beast though! It weighs 65 lbs. However, if you mount it with a helper there is no problem.
When I mounted it to the receiver hitch on the motorhome, I was disappointed to see that the arm that holds the inner bike to the rack wouldn't clear the spare tire... I purchased an 8 inch receiver extender which solved this problem, but introduced a lot of play and bounce on the rack. This problem was solved with a Roadmaster Quiet Hitch anti rattle ring.
One note on the Escapee rack: the trays that hold the wheels are not adjustable as many are for these kind of racks from other manufacturers. This did cause a problem at first, as the handlebar from one bike hit the seat on the other bike. However, since the seat has a quick release leaver, I simply marked the location on the seat post and then dropped the seat down to clear. The bikes are mounted very solidly, with no bounce and I am very satisfied and looking forward to bringing our bikes with us when we go to Maine this summer.