Since you have posted that you will be entering Canada around May 22, snow should generally no longer be an issue. however, you never know, especially in British Columbia and on high passes. Just a heads up for you about crossing the border into Canada at that time in May if you decide to be a few days early. The weekend of May 17 to 19 is a long weekend, and line-ups at the border on May 19 and possibly May 20 will be long, as in hours. Since it appears you are going to Vancouver Island (Campbell River) there is a ferry from Washington State at Anacortes I believe, that will take you over to Victoria, then you can head on up to Campbell River. You can then cross to Vancouver from Nanaimo to get back to the mainland. Reservations if you can make them might be recommended if you are sure of dates.
Sidehillsoup's suggested route is an excellent one, which I would second. Another alternative is to follow Highway 3 at Hope to Osoyoos in the Okanagan Valley, then tour north through the Okanagan Valley to Trans-Canada #1 and then follow it to Banff. Here are some website addresses to the B. C. Provincial Parks and national park websites, which will give you lots of information,as well as what campsites are available along the way. Generally, until the end of June you should be good without reservations.
http://www.env.gov.bc.ca/bcparks/
http://www.pc.gc.ca/pn-np/ab/banff/index.aspx
http://www.pc.gc.ca/eng/pn-np/ab/jasper/index.aspx
Hope this helps. Have a great vacation.