Betsy, if you're coming from Salmon, it's better to take ID-28, then straight-shot it to Rexburg. Going via Challis and down US-93, then cutting back North via ID-22/33 seems to me to be a waste of time, IMHO. ID-28 is very lightly travelled.
As for the Scenic Byway out of Ashton on ID-47, it's nice but with your rig, it would not be advisable if you want to get out and see Mesa Falls. The parking lot and (possible) congestion would be problematic for the size of rig you are pulling. If you think you will want to park the rig and just walk down to the viewing area, again, I wouldn't advise it. The road is narrow in some spots and you definitely don't want to park it along the highway. As an aside, once you turn East out of Ashton, you will have excellent views of the Tetons along this road. Also, there is access to a little slice of Yellowstone Park in the extreme Southwest corner from this road leading to the Scenic Byway, called Bechler Ranger Station (and that's all it is, nothing more), and Cave Falls. Now, the road isn't paved, at least it wasn't the last time I was out there years ago. Bring your mosquito repellent if you plan on going there as they are thick out there for some reason. Also, bring your Bear Spray because it's definitely Grizzly country.
Before you start your climb into the Scenic Byway, there is a very, very nice campground in the Targhee National Forest at Warm River. They completely redid this campground years ago and I must say that it is one of the nicest NF campgrounds I've seen.
Going from Ashton to Jackson via the Teton Valley is beautiful, but you'll end up having to go over Teton Pass. I know that there are a lot of people here who would have no problem with it, but I do. No way I would stress my vehicle pulling that 5er up the pass and then DOWN again, stressing the tranny and brakes, but that's just me. For my part, I would go the roundabout way on ID-31 over Pine Creek Pass, through Swan Valley, ID, to Alpine, WY, then up US-26 to approach Jackson from the South. 2 years ago I happened upon a guy pulling a trailer whose Tow Vehicle must have catastrophically failed because he was at a DEAD STOP on a blind curve going up Teton Pass. I say Catastrophically because I'm sure he would have pulled over if he had had the means at all, but he completely blocked the lane in a very bad spot. This is not a pleasant scenario on your vacation and I couldn't help but feel that he had really made a BAD decision trying to drive the Teton Pass with trailer in tow.
Just my 2 cents. Hope this helps you and others on route planning. You can message me if you want further info.