May I weigh in on Going to the Sun Road?

I have cycled GTTS lots of times - both directions. One can never tire of the experience.
With its even 8% grade, it's really not that bad in either direction.

For multiple reasons, I prefer east-to-west - but it's not absolute.
#1 - I love having the morning light behind me as I ride into the mountains.
The mountains and lake absolutely sparkle.
#2 - Eastside climbing doesn't have the 11:00a cutoff - I like not feeling pressured.
#3 - Although I like to start early - 7:30ish - traffic increases later in the morning.
The longer westside climb means that traffic is significantly more noticeable before reaching the top.
#4 - And, yes, it's 1400 fewer feet - although somewhere you ultimately have to make them up.

Still, if your tour makes west-to-east more reasonable - then do it.
If you ride during the months the shuttle operates, then you can shuttle your panniers up early -
Take the shuttle back down - then ride up unencumbered - even if you are morally suspect.

Speaking of campgrounds -
On the west side:
In addition to Avalanche there is Sprague Creek right next to services at McDonald Lake Lodge - just a bit further.
On the east side:
Rising Sun is WAY, WAY better than St. Mary - plus there are services including pay showers.

Nota Bene!
Visiting Glacier without going to Many Glacier is like going to the Grand Canyon and missing the canyon.
Yes, it is an out and back trip - which many tourers seem to despise - but it is a lovely ride.
All services. This is an ideal location from which to hike into the backcountry.
Also, consider Two Medicine if you are in that corner.