You didn’t say if you are pulling your 5th wheel or not.
The route you described from Jackson into Idaho (22, 33 to 20) goes over 8429’ Teton Pass between Jackson and Victor, ID. It is 10% grade for about 5 miles each side with a couple of 20-25 mph turns thrown in just for thrills! OK for cars and Pickups but for RVs I wouldn’t do it. Occasionally we see somebody in a RV with smoking brakes going 5 mph and a look of PANIC on their faces!
The other choice would be taking 89 and 26 from Jackson to Ririe, ID joining 20 at Rigby, ID. Beautiful scenery along Palisades Res. but the long way to W. Y’stone, MT.
I think that you should just tighten your belt and go north from Jackson to Y’stone NP, West Thumb, Old Faithfull, Madison Junction to W. Y’stone, MT. You can’t beat the scenery with a side benefit of seeing Old Faithfull.
Whatever time you think that you will need to see Y’stone NP you better double it!
Remember that Y’stone Park itself is about 50 miles E/W and 70 miles N/S (2.2 Mil Acres) and the Grand Loop (figure 8) road is about 140 miles around. With a 45mph speed limit (radar controlled), all the thermal attractions, the Bison think that they own the road and will sometimes back up traffic for 1/4 mile or more and the altitude (Canyon Area is 7918 ft, Old Faithful is 7365 ft and Mammoth Area is ONLY 6239 ft) it will take a full day for each loop. When walking take it easy, carry/drink lots of water as it can be very exhausting, then you will only see the highlights. If your luck is like mine I guarantee that Old Faithful will have gone off just minutes before you arrived and will have to wait about another hour and 10-15 minutes for it to go off again kill time by walking around the upper geyser basin. When you are at Old faithful be sure to go into the Old Faithful Inn and look up when inside. When in the visitor center, Inn, Snowlodge etc they will have a sign displaying approximately when the next eruption will occur (+ or – about 10/15 minutes)
We have lived about 110 miles from the West Y’stone entrance for 50 years and go there a couple of times each summer and I still haven’t seen everything yet.