Traveling to Japan in September

I’ve got a trip planned to Japan from August 30th – September 13th. We’re going to do the golden route (Tokyo, Kyoto, Osaka) and had some questions about the weather.

I’ve been getting mixed information so figured I’d ask here. Is Japan really hot and rainy in early September? Our dates are not very flexible but if it’s going to be very bad, I’d consider changing the dates.

Also if we were to change to later dates, would you recommend October or November?

Thanks in advance!