Late May/early June might not be as wet as you are fearing.
We spent about 3 weeks last May all around Kyushu (Japan's large southern island) and the weather was about as perfect as one could hope for. We only got 1 or 2 days of rain in that entire time we were there (one day was steady rain all day, a bit of a soaker; the other day was light rain in the morning, off an on for a few hours, which didn't impact us much as we were driving some distance that day, it did cause some fog and low clouds that made the scenery even more moody and evocative). Otherwise, we had not a single drop of precip in 20 days. Days were almost always sunny, warm and pleasant, never muggy and humid. Nights were cool and pleasant, a bit chilly occasionally when we were up in the mountains. We were there from May 5 - 24, and did a road trip all the way around Kyushu, starting (and ending) in Fukuoka, going around Kyushu clockwise.
Weather is of course impossible to predict accurately far in advance, so no guarantees, but we had great luck with the weather.
Japan is always great - have fun!