I drove that stretch on 89 a couple of years ago. Seemed like a decent road, though I can't account for shoulder width in all areas. Typically a US highway is supposed to have a substantial one.
One place you might encounter some difficulty is the stretch through Logan canyon. I know there is not much shoulder through there, but compared to the rest of your trip it is not a long stretch. Once you get to the Bear Lake area, it opens up and I would say it would be an excellent route because it's not that busy.
One thing I do remember is lots of road construction.....particularly in Wyoming. Not sure of the current status, so I imagine you will check on it. Even though the relative speeds diminish in the construction areas (or they are supposed to!), there is often no space for bikes around the barrels/cones.
Oh, and one more thing. That area will be very high in bugs......not just for when you are riding for hitting your face, but also when you stop the mosquitoes are wicked!