Going to the Sun Road is one of the finest rides in the world.
Yes, it involves climbing, but the road was engineered with a constant 8% grade on purpose.
Way easier than the climb to Rainy Pass.
And yes, there are time limitations for cyclists - along Lake MacDonald and on the ascent.
It's 21 miles from the lodge to the pass - 10 miles gentle and 11 climbing.
You could make it in 3 hours at a steady pace, 3 1/2 hours banana breaks, 4 hours easy.
Not to mention that the uphill climb has jaw-dropping, right-on-the-edge views.
It does mean leaving early - or you can hike up to Avalanche Lake - and ride up after 4.
It doesn't get dark until 10 in late June.
Or you can have the shuttle drive take you panniers up to Logan Pass
or even across if you lodge on the east side, as well.
Then you can zoom up with less weight.
(The driver can leave your bags in bear boxes on top if you are comfortable with that.)
You'll be missing the numero uno section if you skip Going to the Sun.
+1 on all this. It's not as daunting as you might think. Nowhere near as taxing as Rainy/Washington. The above description from the Lodge is accurate. The first 10 miles will not really seem like climbing. When I crossed the first time, two members of our group had their gear shuttled to the top. As noted, you can start before 6 a.m., well before there is any traffic. And what traffic you encounter will be moving relatively slowly. It's not the sort of road where you can speed. Also, there are pulll-offs along the way where you can take a break and let traffic pass.
I will add that if you are planning to go to Missoula and get on the TransAm, Lost Trail and Chief Joseph Passes (three miles apart) are tougher. If your plan is to go to Wisdom, you will have another 25 miles from Cheif Joseph Pass. About 5 miles of that is steep descent. The rest requires pedalling to one degree or another. Between Sula and Wisdom, a distance of almost 40 miles, there is no food or drink except for a rest stop in ID, just after Lost Trail and before the final push to Chief Joseph. Then how do you plan to get back north to rejoin the route? (I can give you an option to Butte.) Staying on the Trans Am beyond Wisdom will get you ever more mountain passes.
If you really want to skip Logan Pass, take Marias Pass and then go to Cut Bank, where you will hit the flats. The climb to Marias is long and there is more traffic, including RVs and trucks, which you won't find on GTS. You can break it up into two days if you can get a room at the Isaak Walton Inn. There is also a private campground before the pass and maybe other motels.
In an effort to tempt you, the last 24 in this set were taken during a 2009 ride up and back down the west side the day the pass opened:http://www.flickr.com/photos/davez2007/sets/72157620763740044/
At a minimum, I suggest you spend a few days at the lodge and make the ride up and back down without gear.
If you do opt for Loan Pass, send me a PM if you want a routing option to bypass Chif Mountain Hwy. and the section in Canada, although I think Waterton Village is worth the effort.