Hey Paul:
I am considering myself a Banff-Jasper Highway expert as I used to live 35 miles South of Jasper. (Sunwapta Falls)
I am actually planning a trip last week of April, and the highway will be clear but will caution that a blizzard can hit with snow accumulation as late as May 24th. This can also occur on the two passes. It is rare however.
I do recall filling up a California families car in early August in a very heavy snow squal, having them inquire as to why I live here.
But in reality, June is perfect as you miss a lot of the tourist traffic. Do not go in July & August, as the highway does get very busy. All of the lodges on the highway target May 23rd (Victoria day Holiday) as their opening day. It is shall we say the start of summer then, with most of the snow gone at higher altitudes.
You should try to hook up with the STN ride in Sept!
I have lots of Banff-Jasper Highway picts on my site.
Bazza