Late May is too early. Look up historic opening dates for Going to the Sun Road. A little googling yielded someone saying the average opening date for the last 8 years is June 23. http://glacier.nationalparkschat.com...php?f=1&t=5998

As a resident of a much further south portion of the rocky mountains, I'm going to say mid-June is about as early as you want to tour in the rockies in the USA. July and August are better. May is a crap shoot. This year, on the weekend of May 22-23, I bailed on a century in Buena Vista CO because it was snowing, and went to Moab, UT instead... and driving back home over the mountains we were in a blizzard, yes a blizzard. In CO, not MT.

In other comments....

Consider that loop clockwise, to take advantage of prevailing west winds on Hwy 3 and more likely south winds on Hwy 93.

The glacier part is fantastic. There is a daily bicycle road closure on Going to the Sun Road - I don't remember the details, but basically you have to be done with the west side and over the pass by (i think) 11:00am. Check with the park service. (http://www.nps.gov/glac/planyourvisit/gttsrfaq.htm)

Adventure Cycling maps (Great Parks NOrth) have a dirt detour that gets you off Hwy 2 from Columbia Falls to West Glacier - i took it, and though the road surface was a bit dicey with 28mm road tires (some loose gravel) it was nice to be away from the cars.

Hwy 3 (Crowsnest Pass) carries quite a bit of traffic, and though it does have shoulders (at least, the other way, which is the way I went) it is pretty busy. There is a wind farm (west wind) at Pincher Creek. You can guess what that means. I did however have tailwind going north up to there from waterton, so it might be a toss-up on where you want your headwind.

There's a nice provincial park campground in Fernie.