You are not going to save money by doing this.
You are essentially traveling Kyushu to Hakodate for the cost of a 7 day JR pass. In green car it appears to cost nearly USD 400 per person. Traveling around Hokkaido is not going to cost you very much by train and I would strongly suggest driving on Hokkaido.
I suggest flying to Hakodate (or Chitose) and getting a rental car or taking trains on Hokkaido.