I'm going on a trip to Kumamoto with my girlfriend. Can you recommend a high-class Ryokan (Japanese inn) we can take a hot spring bath and have lunch included?

I'm planning a trip to Kumamoto with my girlfriend. We've decided to go to Kumamoto Castle, but we also want to take a dip in hot spring somewhere. Kurokawa Onsen is famous, but it doesn't have to be Kurokawa Onsen. Can you tell me a high-end hot spring inn Kumamoto Prefecture where we can take a dip in hot spring with lunch included? Our budget is 20,000 yen per person. We'll go by car.