I have never ridden a bike on those roads, but I have driven them. Going To The Sun Road would be spooky on a bike - especially on a weekend. Absolutely gorgeous country, but often lots of tourist traffic. If you could time it to be able so you could ride it before it was open to cars, but clear of snow, it'd be epic! (Cold too!)

Looks like it typically opens mid June. http://www.nps.gov/glac/planyourvisit/gttsrfaq.htm Be sure and post a trip report with pictures if you do it!

Found a video that will give you an idea of how much shoulder you have:
http://www.youtube.com/watch?v=r5s8tS-ZqCs