Barak -
From what you have posted, I can surmise that you do not know the West very well.
I have lived in Jackson and have cycled the Tetons and Yellowstone as early and as late as possible.
When you do catch those special times when the roads remain closed to cars, it is wonderful.
But it varies from year to year and some years the snows come early or stay late.
With the added factor of Covid, I doubt the parks will have s special cycling period.
I did my first X-USA tour from west to east starting on September 1 and started in Astoria,
but I did a more direct route, left the Rockies in Wyoming, and ended further south in North Carolina.
The TransAm is doable if you start on August 15 and keep up a moderate pace.
(I'm not usre abour you touring experience or what your average day might be.)
It is possible to do a modified route - like the one above - and start no later than early Sept.
That said - this year there are major fires across the Northwest.
Already, there are multiple detours on the TransAm because of fires.
And we're not even into August, yet.
Even where there is no immediate fire danger, heavy smoke is an issue.
I would like to think that the 2021 heat wave might let up in August -
but it is probably more realistic to see it continuing with record heat & fires.
August/September is, after all, the normal fire season in the West.
With a later start and sticking to Adventure Cycling routes -
I would suggest starting in Astoria and taking the Lewis & Clark to Missoula -
Then taking the new Parks, Peaks & Prairies from Yellowstone to central South Dakota -
Then back on the Lewis & Clark to the town of Washington or to St. Louis.
From the Katy you can connect to the TransAM on a short Great Rivers Segment.
Or you can head all the way into St. Louis and the Arch - then continue down the Illinois side.
At that point, you can stay on the TransAm thru Kentucky to Yorktown, Virginia.
Provided that you do in in about 10 weeks - i.e. from Sept 1 to Nov 10.
Climate change has warmed things up in the fall in the East a little more.
We still get some big storms early in the West.
That's why I think it best to take an east turn out of the Rockies by about Sept. 21.
You'll still have some chilly mornings in the Bighorns and Black Hills -
But you'll have great fall weather across most of the rest of the U.S.
Best - Jama