It seems interesting that "less tourists" would be a selling point for Japan, as I can't ever remember thinking there were more or less tourists at any time of the year, simply because Japan is not really a tourist-friendly or tourist-heavy country. (That is compared to places like Hawaii, where you can see a noticeable difference in the volume of tourists.)
I Would suggest July and August are usually very hot and humid on the mainland in Japan. If you are visiting Hokkaido, these months are the best, because the weather is cool and dry there.
I'm going from 29th May to 13th June. I understand it's rainy, but how bad does it get? I didn't want to go when it's TOO hot, I'd rather a little bit of rain. (When it's too hot it makes you tired quicker.)