How late is "late June"?
https://www.nps.gov/glac/learn/news/upload/Logan-Pass-Open-Close-Dates_Press-Kit_10-21-20-2.pdfEast of G.F. I strongly suggest riding via Belt and Hwys 331 & 228.
Very scenic and you miss a good deal of traffic until just before the air force base.
There is a service road running along the west side of I-15 from G.F. to Vaughn.
Nothing to write home to Mom about, but beats riding on the interstate.
Just as you head north from Fairfield on US 89 you have the first kick-ass views of the Rockies.
You are on a bluff overlooking the Freezeout Lakes - give yourself time to enjoy.
Then you have great mountain views all the way to Glacier on US 89.
One option for Glacier to to ride north from Browning to Babb and then into Many Glacier.
Many Glacier has the ginest hiking the in the park, hiker/biker camping, campstore, showers, cafe.
It's 14 miles mostly downhill miles from Many Glacier to Babb, then 9 miles on US 89 to St Mary.
Plus, there is major construction on the Many Glacier Road with significant delays and gravel.
Could they make it any harder on cyclists??
I don't know what the night construction schedule will be - it's not posted.
However, if you leave at 4:00a - bare twilight - with flashers you should get to Babb by dawn.
Still riding with flashers on US 89, you would get to St. Mary by 6:00a.
That would allow you to ride Going to the Sun up to Logan Pass before it gets too busy.
Then spend the day hiking on Highline Traill. (Unless you have acrophobia)
Pic - Lake Josephine at Many Glacier