Both Canmore and Calgary would very likely also have bicycles for rent. The whole 'touring the mountains' thing is big business in those areas.
A person could fly into Calgary and check around there for bicycle rentals. If a person managed to find a suitable one there, it is not a long trip out Hwy 1A (or even Hwy 1) to the mountains ... and Calgary to Banff is 126 kms. The route there is hilly, but nothing overwhelming. From Banff, a person could take the Bow Valley Parkway to Lake Louise and, if a person were feeling particularly ambitious, a person could cycle up to the lake ... and then possibly hike to the Tea House there!!
Or a person could do a ride around Lake Minnewanka. Or head out toward Radium, or Field, or Golden. Or cycle right up the Icefield Parkway to Jasper. So many options, so little time!!
Or a person could fly into Calgary and take the Greyhound (or possibly a shuttle van) out to Banff. I believe there are several such shuttles, like this one:
http://www.banffairportshuttle.com/ , this one:
http://www.banffairporter.com/ , or this one:
http://www.rockymountainskyshuttle.com/ ... a Google search will give you a whole list of them. Then a person could rent a bicycle in Banff ... and go from there.
I've been fortunate for the last two years ... I live right next to the mountains now, and have had the opportunity to ride the roads I've just mentioned -- some of them several times.