I have been over the Bighorns. Route 14 out of Sheridan IS NOT the way to go with an RV. It is very steep going up and even worse going down through Shell canyon on the west side. Route 16 out of Buffalo would be my choice. Still steep but not nearly as much as Rt. 14. A further drive getting into Cody but worth every bit of it.
Since you are in the area, the short drive up I90 from northern Wyoming to Custer battle field. It is one of the best side trips we ever took. Tell you kids about it before you go and they will appreciate it more.
I don't know if it was mentioned but Mount Rushmore is a must see. At the bottom of the mountain there is a town called Keystone that you and the kids will love.
There is nothing like South Dakota,Mont., and Wyoming. You will go back.