We spent 3 wonderful weeks in Kyushu this past May (May 5 - 24, 2024). The weather was delightful, in fact perfect, except for one day when it rained hard all day and all night. Otherwise, we enjoyed clear sunshine and warm (but not hot) temperatures pretty much every day, evenings were cool and pleasant. Kyushu is awesome, we drove all around the island and we had a blast everywhere we went. I would expect more humidity in late June. If the only time you can ever go is late June, then I'd say go when you can.
FWIW back in 2017 we did a trip summer road trip to Hokkaido in late June, and that was absolutely lovely, too - we had a couple days in Tokyo on our way back (first few days of July) and the heat/humidity there completely kicked my butt. Hokkaido is where the Japanese go for summer getaways and it's easy to see why. It's great.