Weather is not the only consideration when talking about getting to East Glacier if you plan to go over Logan Pass rather than Marias. There is an ongoing project to rehab Going to the Sun Road. The last several years, the NPS has been closing the road for extended periods in September to permit work to go on uninterrupted. Depending on the schedule for the year you plan to ride it, you may get shut out. For example, the park's web site notes this:
"This year road construction in September 2012 will make a portion of the Going-to-the-Sun Road inaccessible. The last day to drive to Logan Pass from the west side from West Glacier is September 16. Beginning on September 17th, the section of the Going-to-the-Sun Road between Avalanche and Logan Pass will be inaccessible to vehicles to accommodate accelerated road rehabilitation. There will be vehicle access to Logan Pass from the east side of the park in St. Mary until October 14, weather permitting."
As a frame of reference, the two times I did the western most section of the NT I started in Seattle and joined the route just east of Anacortes on day 3, staying at Bay View State Park. In total, I took 18 days to get to W. Glacier. That included rest days at Winthrop, WA and Sand Point, ID and a short day from Rockport, WA to Colonial Creek Campground just before the climb over the Cascades starts in earnest.
If you are looking for historic weather averages, try here:http://www.intellicast.com/Local/History.aspx?location=USWA9670
But understand that you could get snowed on the mountains.
On the eastern end, you could get some chilly, wet weather in the Adirondaks. Got just that finishing the NT in August. I am pretty sure I would prefer September/early October in OH, IN, IL and IA to the searing heat and massive humidity we had during the height of the summer.