Hello Pierre,
Welcome to the ACA Forums!
I will stick to my native language since you seem so fluent in it. Like Jamawani, I think August could be fairly hot depending on where you go. Coming from France, you would suffer in the 30-35 celsius temps in parts of the western USA.
Additionally, while a month sounds like a long time, in reality, you have a maximum of 26 days being able to ride due to flying and/or pre/post flight bike assembly/disassembly. Additionally, if you factor in a few sight seeing days and/or bad weather days, you could easily get down to 21 days of riding. Assuming your 50 mile/80km per day, you could ride a max of 1,000 miles or 1600km.
With that in mind, I would also suggest something outside your preferred routes. Have you considered flying into Edmonton, Alberta, Canada, take a train or rent a car to Jasper and head south to Missoula, Montana, along the stunning Icefields Highway allowing you a leisurely with plenty of time for biking, sight-seeing, etc. You would use the Great Parks North maps. Be sure to go on the east side of the loop through Waterton, Many Lakes, and Glacier National Parks.
You could take fly home from Missoula. You could also take a train from Whitefish, Montana, near the end of route, to Spokane, Portland, or Minneapolis, or Chicago to potentially less expensive airfares. Since you have more time, you could even take the train down to San Francisco or Los Angeles and get a non-stop flight home. Warning, the trains are not nearly as on-time as they are in Europe.
If you prefer to stay in the USA, I would suggest you do a modified Spokane to Missoula to Yellowstone area. You could fly out of West Yellowstone, MT, Jackson, WY, or Salt Lake City (take a shuttle there). I could help with a route that does the Spokane to Missoula route along bicycle paths and along quiet riverside roads with a few miles of major roads. One of the highlights would be the Route of the Hiawatha trail
https://www.ridethehiawatha.com/
From Missoula, you would use TransAm Map #4 and you don't really need a map for the Yellowstone/Teton Parks.
A longer modified version of the first option would be to go from Jasper to Missoula to Spokane.
If you want to stick to one of your suggested routes, I would suggest flying into Portland and riding from Astoria, OR, to San Luis Obisbo, CA, as this a probably the best, at least in my opinion, part of the route. The part between Monterrey, CA and San Simeon is a definite highlight. Just do NOT ride on a weekend as it is crazy with motorcycles.
A final suggestion and one that many Europeans (and North Americans since we do not get as much vacation time) do is to ride the TransAm in segments, i.e. one month at a time over a number of years. The TransAm really is a very good ride.
Since you do not say if you will depart France on August 1st or August 20th, that could factor in somewhat to the recommendations given.
Here are links to the national (not federal) bus line:
https://www.greyhound.com/a regional bus line that shuttles bikes without boxing:
https://saltlakeexpress.com/Salt Lake Express will also take you to Salt Lake City Airport if that is easier but then getting a box may be a pain at the airport.
and nations train service:
https://www.amtrak.com/homeOn the train service only certain stations allow bicycles and some only allow them if the bicycle is boxed. AmTrak bicycle boxes are huge, just turn the handlebars, remove pedals, and lower the seat.
When boxing the bike, go to a local hardware store and buy some HVAC insulation tubing to protect the frame. Very inexpensive protection.
Whatever you choose, I wish you a wonderful trip! Tailwinds, John
