I've done part of your route in reverse -- Tetons/Yellowstone to Glacier to Anacortes. It should be doable. Beware of the (in)famous "5 passes in 4 days" near the beginning of the NT (FWIW, Rainy and Washington Passes are so close they shouldn't count as two). Those "two" can make for a long day if you're motelling, although there's campgrounds between Concrete and Mazama.
The worst grade you'll hit on this section of your route is about 8% for 0.8 miles, but there's lots of 6% to make up for the shortness of that climb.
Make sure you have low gears for your knee's sake; I'd shoot for 20 gear inches or lower. Road-group gearing at 34-28 or even 30-34 lows won't cut it.
You'll probably be limited on start time by the opening of the Washington state route 20 (Northern Cascades) passes. Google for Washington DOT pass closure to get more information. IIRC, their average opening date is mid-May, but sometimes it's closed into June. They closed that road a week before it was planned to close this year, but you'll have to wait until next March to see the late season snowfall before you make any guesses on opening date.