Hi there,
I think it should certainly be doable but what I am curious about
is if you already made a plan of how to travel from Kumamoto to Daisetsuzan Nartional park.
Kumamoto is all in the south and Daisetsuzan in the North of Japan.
Here is a little estimation of how many days you could spend in each place and how to travel around.
Day 1 arrival
Day 2 - 4 to explore the city and day trips to Nikko and Kamakura.
Day 5 travel to Hakone
Day 6 - 7 Enjoy the Hakone Area
Day 8 travel to Matsumoto + Night there
Day 9 Morning in Matsumoto than on by Bus to Shirakawago.
Day 10 Shirakawago + travel on to Kanazawa.
Day 11-12 Kanazawa
Day 13 Travel to Kyoto + Kyoto sightseeing
Day 14 Kyoto
Day 15 - 19 Kyoto Sightseeing + Day trips to Osaka, Nara, Hiroshima and Miyajima
Day 20 Travel to Kumamoro
Day 21 Kumamoto
Day 22 Kumamoto + Airplane to Hokkaido for Daisetsuzan
Day 23 - 24 Daisetsuzan
Day 25 Travel to Tokyo by Plane or take the train down with a stop at Hakodate or Aomori.
Day 26 Tokyo or 2nd leg to Tokyo by train.
Day 27 Tokyo + Shopping
Day 28 To Narita Airport for flight back home.
So yes I certainly think this is possible. I think that it makes for a very nice trip! If you don't want to travel all the way up to Hokkaido for Daisetsuzan I would recommend staying a couple days longer down in Kyushu, the area has a lot to offer, great places to visit are Beppu for Onsen, Mount Aso which is a beautiful volcano and Kagoshima which offers a beautiful view on the ocean and Sakura-Jima.
Have a great time in Japan!
-Daniel
