Can't advise whether route 89 the the "best" route, but we did drive that road from Gardiner up to East Glacier the first week of June, 2010. From East Glacier you might want to consider driving north to just before Babb and take route 464 to get to Browning. We used 89 between Browning and East Glacier both directions with our 24ft Class C, no toad, but you might be more comfortable avoiding the curves and short grades on that section of 89 with your longer length. The rest of 89 from Browning to Great Falls was an easy drive. Coming north from Yellowstone, we did an overnight at Great Falls so we could visit the Lewis & Clark Interpretive Center as well as adjacent Giant Springs.
South of about Neihart, route 89 starts a long climb to Kings Hill Pass at 7,400 ft in the Little Belt Mountains followed by a long descent to White Sulphur Springs. From there it's all level in a wide valley all the way to Gardiner.
When we did it there was lots of snow on the mountains and we thought it a very scenic drive.