And in Montana at Glacier NP there is a non-Canada option "Marias Alternate". My 2008 copy of the NT shows that option. You can avoid Canada if you desire and still be following NT maps.
IMO, that's like going to Brooklyn Heights and not taking a walk on the promenade. In '09 we were forced to ride Marias east to west because Logan Pass was still closed. West to east is a long slog up to Marias. The shoulder leaves something to be desired and there can be some traffic. It's not particularly scenic in its own right, and certainly nothing compared to Going to the Sun Road. There are also very few services between W. Glacier & E. Glacier. Aside from the Isaak Walton Inn, I remeber passing only one or two restaurants and one campground. The manager at the later was nice enough to let us bum some water.
If you are dead set on skipping Alberta, I highly recommend at least doing Going to the Sun to St. Mary. From there, there are a couple of ways to get to Cut Bank wihout going into Candada. In '09, we took U.S. 89 south from St. Mary to MT 49 (Looking Glass Hill Rd.) to East Glacier, where there is a hostel, a motel, camping and food. There, you can pick up U.S. 2 east through Browning. This route gives you the option of a side trip into the Two Medicine part of Glacier N.P. And the views, especially from Looking Glass, are pretty amazing. It’s my understanding that Starr School Rd. from U.S. 89 to Browning is easier but not nearly as scenic. Finally, you could just stay on U.S. 89 south from St. Mary to Browning. You can see all this on Google Maps.
Here are some photos from our ’09 trip to/from Whitefish into B.C. and AB:http://www.flickr.com/photos/davez2007/sets/72157620763740044/
The final 26 are from the west side of Going to the Sun up to Logan Pass. The 7 before them are from between St. Mary and E. Glacier. The grazing deer to the general store are from Waterton Village to St. Mary via Chief Mountain.
This may all be an academic exercise depending on when you get to W. Glacier. They just had another big snowfall the other day. Plowing of Going to the Sun is going very slowly. I would be surprised if the road opens by the middle of June. You may be forced to go over Marias.
While it was some time ago, I did the entire NT (’99) and the portion from Anacortes (actually started in Seattle) to Glacier the following year. Send me a private message if you have any specific questions I may be able to help you with.