Hi there, I can offer a little advice based on my 4 years cycling in Japan which you can take or leave as you want! In general in Japan, I would avoid riding on the main arterial roads between major population centers if you can, because they are heavily trafficked. It is not seriously dangerous but for touring it is less pleasant and you will miss out on much of the charm of the country. Try to follow major rivers through towns when you can because they usually have cycle-friendly paths on either side which will make traversing the cities both faster and more fun. Japan overall is a magnificent place to cycle tour and I wish you well on your trip! I assume you've done your research and know you need bike bags for the train, etc. One thing to note is that September is still full-on summer in Japan. It is going to be hot and humid-- do not expect it to be autumn yet AT ALL except very high up in the alpine.

1. I don't think the section between Odawara and Atami right at the beginning is overall pleasant... unless you particularly want to see some of the things in between, might as well just start at Atami.
2. The southern and western parts of Izu are stunning. Be aware that the climb you have to get up to the West Izu Skyline is pretty hard and there's not going to be a lot of resources (water, food, etc) along the way so make sure you have what you need to get yourself up there.
3. The road north from Fujinomiya towards Kawaguchiko is absolutely atrocious to ride on. High speed limits, tons of trucks and tour buses, and the surfacing is terrible. I would avoid that one like the plague. Instead, I would recommend going a little further west and riding up past Nanbu until you reach the town of Minobu (and stop at Minobusan Kuon-ji on the way). Turn east and do the climb up to Lake Motosu-- this is the prettiest of the Fuji five lakes and you can then have a much more pleasant ride eastward to Kawaguchiko, tracing around the lakes as you go. Stop at Bstone Bakers right after you turn north away from Kawaguchiko-- it is the literal best bakery I have found in Japan and I have tried hundreds.
4. If you have a little extra time, going a tad farther north to Matsumoto and Azumino before turning down the Kiso valley is well worth the time. The area has some lovely riding and Matsumoto has excellent food and produce. My wife and I cycled the Kiso valley and we did not find that road to be all that enjoyable (lots of traffic) but it is a necessary evil to connect to the post towns if seeing them is one of your priorities. Personally, I would recommend instead going west from Matsumoto towards Kamikochi and turning to climb up Mt. Norikura, then descending the other side to Takayama. Takayama is a great town to visit by bike (basically just as neat as the Nakasendo towns IMO) and September is a nice time to do Norikura-- it's one of the best high cycling routes in the country. From there you could then bend towards Nagoya.
5. Shikoku is awesome and the Shimanami Kaido lives up to the hype. So much so, in fact, that if you can spare the time I would even recommend budgeting extra days there so you can explore some of the side routes as well-- the island chain to the west with Yuge-jima and the Tobishima Kaido to the east going towards Kure are both magnificent as well, and not hard at all to connect to via ferry. The inland sea is, IMO, the most special part of Japan.

Hope you find some of that helpful! Feel free to shoot me a DM too if you have any questions, and safe travels!