I think at that time of the year, the weather should be OK regardless of the direction you travel (although as stated by others, you can hit unexpected inclement weather in both the Rockies and coastal mountains). I also think it matters little in the Rockies as to when you travel during that time frame as it relates to crowds. Banff et al are crowded all the time.
Having said that, I would suggest you start in the Vancouver area (a good rv park to stay at is Capilano River RV Park - on the "North Shore" just across the river from Stanley Park and downtown Vancouver and within a few minutes drive to three local mountains - Grouse Mtn, Cypress Mtn, Seymour Mtn - all hiker and biker friendly).
From there head north to Whistler, another hub for great hiking and biking (you have to do the Joffre Lake chain - three spectacular lakes, the farthest about a 4 kilometre hike - very recommended - try googling them to see the views - trails are steep but in real good condition). The Area needs a couple days to take in all the activities.
Continue north, then east towards Kamloops/Okanagan area and dry almost desert like/wine country. A lot of activities in this area as well.
Then continue eastward through the Revelstoke and Golden areas (again more hiking/biking) en route to Banff (sounds like you know this area so no need to provide activity advise).
Return to the U.S. via Highway 2 from Calgary Alta.
Although I drive (and pull a trailer) a few times a year from Vancouver BC to Denver Co via the main interstates, I have also taken various alternate routes and can give suggestions on these routes if you wish (like through Crater Lake area and spending time in the Columbia River Gorge doing hiking etc.).
Regards,
Marv