Nice scenery along that route, at least once you're past Bozeman. (You may be driving thru parts of town there.) Sorry, I don't know about construction; there's usually some somewhere in this area during the summer, since there's such a short season for it. Grizzly is a great CG choice, IMO.
While along I-90, I hope you have time to stop at the Little Bighorn National Battlefield for a day (2 decent CGs nearby for an overnight). It's well worth the stop. And of course Devils Tower is off I-90 and worth a stop as well. The KOA there is right at the base & has great views of the tower; there's also a no-hookups NPS CG inside the park.
Have a wonderful time in Yellowstone!