The Canadian national parks have fire trails which permit you to access backcountry campsites on a bicycle.
This is something that U.S. national parks do not allow for the most part.
So, depending on what you are riding, it is certainly worth considering.
But I would strongly suggest delaying until, at least, mid June - maybe July 1.
(Another option is to consider south to north - still, before June 15 is tricky.)
The major campgrounds in the Rocky Mountain national parks tend to be zoos.
That's why I do backcountry camping or seek out small, isolated campgrounds.
Plus, in the Canadian parks there are rustic hostels every 25 miles or so.http://www.hihostels.ca/1084/Home/Canada-Hostels/Alberta-Hostels/index.hostel?p=ab
I used to live in Jackson and have ridden the parks up to Jasper many times.
Even though it is off your planned route - Mount Robson Provincial Park is amazing.
Plus you can ride into a backcountry site there, too.
Riding back from Sunwapta Falls along the Athabasca River you get to an incredible horseshoe.
360-degree panorama - and, usually, one or two other people.http://www.canadianrockies.net/downloads/jnpbackcountrymap.pdf
Similarly, in Banff you have backcountry bikeable sites on Red Earth Creek and Brewster Creek.
The Spray River Trail from Banff south is fabulous and has a backcountry site, too.http://www.canadianrockies.net/downloads/backmap_e.pdf
The Spray River Trail connects to the Smith-Dorrien Road to Kananaskis.
It's unpaved, but hard-packed - and so much better than riding thru Canmore. (Done both)
Kananaskis is a secret jewel - visited mostly by Albertans.
At the lake there is a store and the walk-in sites by the lake are stunning.
The Great Parks North map routs you thru Kootenay N.P which is so-so.
I cannot believe they do not go via Kananaskis.
Two ways south of Kananaskis -
1 - Paved via Highwood Pass and Longview
(Or you can do unpaved on the Trunk Road along the Livingstone River)
2 - Unpaved over Elk Pass to Elk Lakes and then via Elkford and Sparwood
15 years ago I was the only person camping in Elk Lakes Provincial Park.
There wasn't even a ranger at the ranger station - amazing, but a little spooky.
Waterton N.P. has a giant campsite in town - and there are some good cafes there.
But there are backcountry sites you can hike into from town just 1.5 miles away.
Many Glacier is not to be missed - great hiking, pizza cafe at the general store, etc.
Along Going to the Sun Road - opt for Sunrise on the east side and either Avalanche or Sprague on the west.
St. Mary and Apgar are zoos - huge RVs and constant noise - plus you pay extra.
Finally, I suggest US 89 on the east side over Hwy 83 on the west side in Montana.
Hwy 83 has more traffic, fewer shoulders, and limited views because of dense forests.
US 89 has been improved over the years with more shoulders and stunning panoramas.
(Of course, the views sometimes do come with winds - that's the trade-off.)
US 89 south of Great Falls is magically quiet with plenty of camping in the Belt Mountains.
The Eastside Road south of Livingston has almost zero traffic south of Pray.
Hot springs at White Sulphur Spgs and Chico.
Plus, you enter Yellowstone via the historic Arch at Gardiner.
Don't let people scare you about Yellowstone - just plan.
If you ride early and late you will encounter little traffic.
Getting up before sunrise and riding super early also lets you see more wildlife.
Then you can explore trails and scenery during midday.
Also, since all the campgrounds have hiker/biker sites, you can ride late without worry.
If you are entering Yellowstone at the North Entrance, (Gardiner)
then I suggest a route via Norris, Canyon, Lake, and West Thumb.
Norris makes a great overnight stop and you can do an out-and-back to Old Faithful.
(Old Faithful is the zooiest of zoos - they even have a cloverleaf interchange.
Norris geyser basin is so much more rewarding - especially early and late.)
Don not miss the Grand Canyon of the Yellowstone and Hayden Valley - the Serengeti of America.
Best campgrounds - Indian Creek, Norris, Canyon, Lewis Lake. Skip Mammoth, Bridge Bay, and Grant.
Lastly, in Grand Teton N.P. there are great hiker/biker sites at Jenny lake close to the camp store.
Do plan on hiking - esp. from String Lake up to Leigh Lake and beyond.
(The Cascade Canyon Loop is brutal and will take all day on a long summer day.)
Plus, there are great backcountry sites on Leigh Lake and up further.
They require a permit, but you can stow you gear in bear boxes at String Lake.
Hope that helps - - J
Pic - Big Bend Campsite in the backcountry of Jasper N.P.