Itinerary 12 days in Kyushu early May

I'm currently planning a 1 month vacation in Japan, during which I'd like to spend 11 (or 12 days, depending on the recommendation) in the Kyushu region, and I would like some advice on my broad itinerary. I won't lie, I'm a bit overwhelmed and need people to guide me a little bit.

May 1st | Flying early from Tokyo, airport to hotel, then spend the day in Fukuoka | sleep Fukuoka

May 2nd | Visit Fukuoka, heard the food scene is amazing, visit shrines, etc. | sleep Fukuoka

May 3rd | Continue to hit touristic and foodie spots in Fukuoka | sleep in Fukuoka

May 4th | Early Shinkansen to Nagasaki, atomic bomb museum, peace park | sleep in Nagasaki

May 5th | Nagasaki, Chinatown, shrines, haven't looked up everything yet | sleep in Nagasaki

May 6th | Nagasaki -> Shimabara for the day and ferry to Kumamoto in the evening | sleep in Kumamoto

May 7th | Kumamoto just for the day, heard the castle is beautiful | sleep in Kumamoto

May 8th | Go early to Kurokawa (bus? car?), go first to Mount Aso maybe in the morning? How's the situation in Kurokawa for a lightly tattooed person? | sleep in Kurokawa

May 9th | Go early to Kagoshima (train? car?), on the way go to Takachiho maybe? and Kamishikimi Kumanoza Shrine? | sleep in Kagoshima

May 10th | Kagoshima, Sakurajima, etc. Food scene look amazing there | sleep in Kagoshima

May 11th | Kagoshima in the morning, then go to Kirishima to be closer to the airport | sleep in Kirishima

May 12th | Kagoshima airport flight for Okinawa

First of all, does that look feasible? Seems ok to me, but I've never been there.

I'd love to rent a car, but does my itinerary needs it? If not, is there a Kyushu Pass that covers everything that I want to do here?

With the possibility of renting a car, what hidden-gems/out of the way village should I add or swap in my itinerary? Should I rent it starting from Kumamoto?

Also, I love craft beers, any recommendation in the area?

Would love your help!

There are many great places to see, though for some, yes you do need a car. For your plans, you need to expand some day itineraries more. Also it looks a bit strange to go all the way down to Kagoshima and drive back up to go to Takachiho. It would also help if you gave more info on your interests.
One of the first things you need to do is decide if you're willing to drive. That can greatly expand or limit your options. For example, seeing Unzen and Shimabara without a car would be very troublesome.
https://goo.gl/Q843Bg

Have a look at the Kyushu rail passes and see what you think. https://www.japan-guide.com/e/e2361_05.html

We spent a week there just using trains and there was plenty to see. If you get a car it does expand your options.

We stayed a few days in Kumamoto and did day trips from there which saved changing hotels so often.

Enjoy your trip!

Looks pretty solid IMO, no need to add anything IMO.
You could rent a car in Kumamoto and return it there, leaving an easy 1 hour train journey to Kagoshima or do a one-way rental returning in Kagoshima (more expensive of course).
Kyushu is really beautiful, I loved driving there, mountains and volcanos everywhere.

Here are two food places in Kagoshima that I loved:
https://maps.app.goo.gl/SrnNu2f4fFpbr2Nb6 (Sushi)
https://maps.app.goo.gl/R7Bgt8GRPCctvs676 (Izakaya specialised in sake, food is so good)