If you could do a three-point plane ticket or rent a car one way - -
I would suggest Grand Teton thru Yellowstone to Glacier -
Ending with the spectacular Going to the Sun Road.

But you didn't say when you were planning on doing this.
The Grand Teton to Glacier ride is best done late June to early August.

You can fly into Jackson, Wyoming pretty easily now.
There are bike shops in town to get you all situated.
Then it's only 15 miles to the hiker/biker campsites at Grand Teton.

Yellowstone is tricky with traffic -
But if you cycle in the early morning and, perhaps, evening -
Then it is actually quite lovely.
I've cycled every mile of pavement in the park.

US 89 makes a fabulous - low-traffic ride north in Montana -
And the scenery is lovely, too.
For the first stretch between Gardner and Livingston -
there is a back road on the east side of the river thru Pray.

Great Falls has a nice riverfront bike trail with all services.
Then US 89 heading north has expansive views of the Front Range.
There are a couple of ways to do Glacier - which is a cyclist's paradise.
But riding Going to the Sun Road is life-changing.
Start early-ish - say 7:30a - so you beat most of the traffic -
Also, you have the morning sun illuminating the Continental Divide peaks.
Spend a little time hiking the Highline Trail up top.
Then scream down the other side.
You can finish in Whitefish or Kalispell and fly/drive out.

https://www.crazyguyonabike.com/doc/..._id=26427&v=O5

It wold be closer to 10 days than 7.