Well, you are prepared to pay USD 379 for a train ticket. I can tell you that it will be less than that. There are also LCCs operating in Japan. Skymark, Starflyer, etc. Many fly to Chitose on Hokkaido.

Skymark flies FUK-CTS non-stop. Also Air Do flies to Hakodate but you would connect at HND. You can fly from a large number of airports on Kyushu to Haneda on ANA. And this would be an ANA ticket since they have part ownership of Air DO.

Ok, Skymark ranges from JPY 12.9k to 18.9k for FUK-CTS depending on advance purchase and availability. The full fare is 24.9k.

Note that domestic tickets in Japan go on sale 60 days in advance and the cheapest advance purchase is 28 days. If you have *A or Oneworld miles then domestic Japan travel is good value on ANA and JAL, respectively.