You can save a lot, but that is still moving at a pretty fast pace and you'd be missing on a lot of good places in the areas.
Hiroshima has a lot, and not just within the city and Miyajima. In addition, there are other numerous great places you can visit in the area. Okayama is not far and it is famous for its castle, plus Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
Hiroshima also has a number of good places. Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
For Fukuoka, don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely FREE!
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here.
In terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, and another day to see areas nearby like Shimabara/Unzen or whatever interests you. Gunkanjima, while it takes 3 hours, will blow your mind.