My suggestion would be to go to Red Lodge and do the Olds equestrian event. Day trip to Drumheller and Tyrell Museum from there (1 1/2 hr each way). Then head north for one day in Edmonton. Head to Jasper (Whistler's), then Banff. You'll end up in the south part of the province on the TransCanada for a straight shot home.
Banff can be very busy, but there is a lot to do there. The Tunnel Mountain Campground is right in town and is ok. No fires in most of it though. Your trailer is too long for some of the other campgrounds close to Banff and I wouldn't recommend Lake Louise for the reasons already mentioned.
Bow Valley PP is just outside of Canmore, right on the Bow River and it is nice too.
Keep in mind that the National Parks and Provincial Parks have different booking dates. Banff reservations start April 2, Jasper starts April 1. All provincial parks are usually 90 days before arrival.