Many people have done the entire GD route on a touring bike. It should be no problem to use one on the short Canadian section between Banff and Sparwood. Further south, on the washboarded roads of Wyoming and the rock-strewn roads of New Mexico, most touring bikers probably have wished a bike with front suspension.
Wider tires are pumped to a lower pressure, which provides two benefits. Firstly, they give more suspension to shocks. Secondly, the contact area with the road surface is bigger and hence the imprint on a soft surface, like gravel, sand or mud, is smaller. So rolling resistance on non-paved roads is less for wider tires. This is reverse on paved roads, although the effect is much weaker . So you have to make up your mind which is more important: the GD section on dirt or the Great Parks on pavement.
I have read several GD blogs from this August, after the flooding in June. It seems that the Goat Creek Trail out of Banff can still be used, although one bridge was wiped out. The road from the Upper Kananaskis Lake to Elk Pass had no problems; about 2 miles are steep and probably require hike-a-bike; the remaining has a gentle gradient.
The main bridge halfway between Elk Pass and Elkford was wiped out. This requires a 30 miles bypass on the east side of the Elk River, with some fording. See pictures in http://greatdivide2013.tumblr.com/
It seems that also the Fording Rd south of Elkford, a dirt road, is damaged and impassible. This is a pity from a scenic point of view, but the bypass on pavement is straightforward and much less strenuous.
In September conditions may have changed again. Keep an eye on the Routes section of this Forum and on GD journals on CrazuGuy: http://www.crazyguyonabike.com/doc/categories/?o=RrzKj&category_id=193&doctype=journal¬