When is generally a good time to start?

Next year might be a good year to get an early start. There is a very strong El Nino brewing which typically means a lot less snow and early spring.

But there's no guarantee about when the weather will change. It is not unusual to have snow in the mountains in June. As long as you have some flexibility, you should be ok.

You shouldn't need reservations in Banff until the end of June unless you are planning to be there on a weekend. The campgrounds will likely be full on weekends from the May long weekend until September.