Yeah this is a bit too general. You say you could fly into either TYO or KIX and are interested in going to Fukuoka. That pretty much leaves open the entire country other than Hokkaido as a possibility. You need to tell us how many days/nights you'll spend in Jpn. At the very least, decide where you'll fly into and out of, and how long you'll be there. It seems like the only thing that can be said definitely so far is that you should stay away from Kyoto if you want to avoid the crowd and not stay there in foliage season if you're on a budget. Interest in nature, culture, history and food doesn't narrow things down much. Lodging like Dormy Inns is available everywhere. Good hiking in Nov can be found just about everywhere whether near Tokyo (Takaosan, Chichibu, Oku-tama, Hakone) or Osaka (Kongo-san, Rokko Mtns). So that doesn't really help dictate your itinerary.

Asian tourists are everywhere these days. In specific regions of Jpn, it is hard to get away from them anywhere you go. Even in places that I thought were off the beaten paths such as Muroran to the north and Shimabara to the south, tons of foreigners. On the other hand, there's plenty of enjoyable rural spots in Saitama and Tokyo-to I've been to recently where I didn't notice any foreigners. However, I'm just not sure a lot of those places are worth it to you. I don't know if anyone's going to be able to give you a comprehensive list of where you won't see foreigners. And if want to go to prominent places that will take your breath away, you're going to have to accept that there'll be other foreigners.