Kumamoto to Aso - car for a night or less?

I'm planning a very hectic 1 week in Kyuushu in July and have now ran into a dilemma of for how long should I rent a car in Kumamoto.

My plan is to stay in Kumamoto for 2 nights, renting a car from Kumamoto station for max. 12h, and do a day trip to Mt. Aso and Takachiho Gorge, stopping in other places that seem remotely interesting. I'd then continue to Kagoshima the next morning from Kumamoto.

However, would you recommend spending the night in Kurosawa onsen and only returning to Kumamoto the next morning? I have also planned to stay in Beppu and I'll be going to another onsen once I reach Tokyo, so I'm not really that keen on adding another onsen to my trip (especially in July). I'm all for changing my mind and please let me know if there are other places in the Mt. Aso area where I could stay.

Thanks!

Kurokawa onsen is about relaxing doing Aso, Takachiho AND Kurokawa in one day seems to be totally impossible.
I would question if Kumamoto , Aso, Takachiho , Kumamoto is even doable in a day. Specially because the attraction in Takachiho is in the evening.

That's about 6 hours in the car for both sights.

Overnighting at Kurokawa will only save about 90 minutes.

Probably still not enough time to do both.

I'm going to stay a day in Kumamoto, so Mt.Aso&surroundings would be a completely separate day trip. So the itinerary looks more like:
- Day 1: Arrival in Kumamoto in the morning, explore the city and sleep in Kumamoto
- Day 2: Mt.Aso and surroundings (spend night in Kurosawa onsen OR Kumamoto?)
- Day 3: Move to Kagoshima

You might be able to theoretically drive to these two main locations, but I would not say that it was be a fun experience.

As a person who has driven these roads - they are slow, windy and often take a lot longer than google maps might show.

Takachiho is worth a good day. Aso is also a nice few hour drive - although there is a bit less to do there. The drive is spectacular though and I actually wanted to drive slower than the speed limit.