A buddy and I did the trip from Jasper through Lake Louise, Banff, and back home to Calgary at the peak of the summer in 2014 (last week of July). We were self-supported and camped in the mountain park campgrounds each night. If you go mid-week, there won't be a problem getting a campsite. Actually, a couple of the ones we stayed at (Jonas Creek and Waterfowl Lakes) the campgrounds were maybe 1/3 full. Just stay away from the larger, serviced campsites that attracts the RV crowd -- those ones can fill up and don't have great sites for people in tents and tend to have a much higher proportion of campsites that are on the reservation system (the smaller campgrounds always have first-come, first-serve sites, particularly for tenters).
Regarding time of year, September can be the absolute best time of year out here, but it is risky. This past year (2014) we had a terrible cold snap and a lot of snow here in the first week of September, and it would not have been a fun time to be biking through the mountains, especially on the east slopes of the Rockies (west slopes on the BC side are much less of a concern).
As for how much time to do the trip, you can easily go between Jasper and Banff in 3 days, even with stops to see some of the tourist attractions, like the Columbia Icefields interpretative site. We probably could have made it all the way home to Calgary in 3 days if we had planned our trip a bit better. I haven't been down to Glacier yet, but having driven that way on both the BC (west slopes) side and Alberta (east slopes) side, it is immensely stunning scenery either way. Both southeastern BC and southwestern Alberta, plus that northern part of Montana below, are all some of the most amazing wilderness you find left in North America.