There are two ways out of Jackson. The pass is steep up and steep down and steep curves. People drive it every day. Been over it many times in a car. It is a little shorter but not worth the the strain on your RV.
Take the canyon out to Alpine turn right, the road will take you right to Idaho Falls. It is a little curvy but a very beautiful drive through the mountains only one straight line grade on wide four lane highway to deal with. When in Idaho Falls get on I-15 to Pocatello, about 60 miles. Just before Pocatello, stay to the right and take I-86 which becomes I-84, same road easy to see turn off. Next stop Burley. This is the easiest route to take with no strain or pain. Distance to Idaho Falls from either the Canyon or the Teton Pass in not a big deal a few miles.