Kyushu 7 days with 2 young children

No need for apologies. I'm very happy there are people like yourself willing to give these suggestions out for us to accept. nothing better than finding a gem that no one else has before, and sharing it with all to enjoy also.

Suginoi is mid-range accommodation in Beppu. It is a sort of spa resort. There're accommodations which has private onsen facility attached to to the room. We stayed at Sawayaka Beppu no Sato last summer.
I visited Kumamoto Castle a few weeks ago. I was with a volunteer tour guide to see the castle tower from the other side of the moat. We walked for about an hour from Josaien to Kato Shrine and on the way back.

Forgot one more question guys. Are there any decent, sandy beaches I can take the kids to? Again, I don't mind driving. Just clean, safe water. :)

Plenty if you are going to the Miyazaki/Nichinan coastline. Some are semi sheltered like Aoshima Beach. But there are little pockets of beach along the whole coastline. Use google maps and you'll find plenty.