Ross -
I live in Wyoming and have been here since 1990.
I live near the Bighorns, now, but was in Jackson in the 1990s.
You will be lucky if you even see a bear. Extra lucky if you see a grizzly.
You won't have a problem riding unless you do remote trails and startle a bear.
The real issue is camping. Make sure you use bear-safe camping practices.
Always store or hang your food and toiletry items.
Never eat in your tent - not in Wyoming, not in Kansas, not in Jersey.
If you have, it may be wise to wash out you tent with unscented soap
and let it dry in the summer sun. (Bears have 2100X the nose you have.)
https://www.fs.usda.gov/visit/know-before-you-go/bearsI have never had a single problem in decades of hiking and cycling.
About Yellowstone.
Teton Pass is closed and, if reopened, will have monster traffic.
The key is to start early and, if needed, do some later riding.
Roads are busiest 9a to 5p.
Since you are westbound I'll give you specifics.
There's fantastic hiker/biker camping at Jenny Lake in Grand Teton NP.
Jenny Lake to Flagg Ranch - 35 mi
You have all day. I'd stop at String Lake and do the short hike up to Leigh Lake.
String Lake has some of the most spectacular vistas.
The park road is slower and has moderate traffic.
When you get to US 89 is will be busier, but with a small shoulder.
Then there's a chunk of a climb to Flagg Ranch.
There's a camp store and restaurant at Flagg Ranch.
I like treating myself to an early dinner there because ...
Flagg Ranch to Lewis Lake - 14 mi, 1000 ft. climb
I like to do this after 6p. The road is steep in places without any shoulder.
There is a lot of traffic leaving the park southbound, but very little northbound.
And I prefer Lewis Lake to Grant Village even with the mosquitos.
Grant Village is huge and noisy - I rarely get much sleep there.
Hiker/biker camping at Lewis Lake.
Lewis Lake to Lake Village - 30 miles
I like to leave at dawn, skipping Grant Village altogether.
I do like to stop at West Thumb and enjoy the geysers when it is empty.
Then I follow the shore of Yellowstone Lake to Lake Village.
If before 8a, light traffic and small shoulders.
Make sure to take Gull Point Road and enjoy a secret, unvisited place.
then there's a rough bike trail along the old roadbed to Lake Village.
(Higher bear risk on the trail - you can also stay on the road.)
There's a camp store and lunch counter,
but the Lake Hotel dining is lovely, if you don't smell too bad.
(You can always do a quick sponge bath in the restroom.)
The views of Yellowstone Lake are stupendous.
You can head over to Fishing Bridge and the visitors center there.
Plus do a few short hikes along the lake shore.
Again, I would wait until after 6p to ride to Canyon.
Fishing Bridge to Canyon - 16 mi
Hayden Valley is the Serengeti of America.
But riding later you will have far less traffic and more wildlife.
The road sucks - narrow, no shoulders, and crumbling edge.
But it is truly worth it. Plus there are hiker/biker sites at Canyon
so you don't have to rush.
Canyon has showers, laundry, a big camp store, restaurants, visitors center.
Plus it has some of the best hiking in North America.
Try to spend two nights if possible.
Canyon to Norris - 12 mi
Moderate climb, small shoulders, then a killer downhill.
Norris Geyser Basin is worth a long stop and a good walk.
So much better than Old Faithful which is a zoo.
There's no water available here, so stock up beforehand.
(You can also ride over to the campground to top off.)
Norris to Madison - 13 miles
Recently rebuilt park road with small shoulders, mostly downhill.
Madison has a nice hiker/biker area - but no store or cafes nearby.
I think it is smart to camp here as camping further west is hard to find.
Set up early and ride without panniers up the Firehole River if you wish.
Again, evening riding is lovely.
Happy Trails! - Jama
Journal from back in 2005 - starting at Jenny Lake
https://www.crazyguyonabike.com/doc/page/?o=3d2&page_id=26429&v=OF