Jun, it is wise to heed to advice about the dangers of Mt. Fuji off season. It is one of the most dangerous mountains in Japan when it is still capped in snow and ice, in great part because it is totally exposed and the winds up there can be ferocious. A very experienced, high-altitude mountain climber friend of mine (Chris White, some of you may know him) almost died up there during winter, when conditions are safer than they are at the moment (snow more stable, less ice, rarely any rain), mainly because the winds were so powerful he could hardly move. Going up there during the rainy season in Japan is most definitely not a good idea. The rainy season is torrential rain, not just any old rain. Zero visibility due to rain and clouds. And that coupled with very unstable snow and lots of ice... not a safe time to climb.
If you insist anyway, and you are on Facebook, look for the FB group "Hiking In Japan", ask to join, and then pose your question there. There are some extremely experienced Japan mountaineers there, including Chris White, who will be happy to provide you with the technical knowledge you need. I suspect, though, that most of them will strongly discourage you.
I'd say, wait at least until mid-July, when you can be sure of more stable weather and far less snow. This year the snows were very deep, so it might take longer than normal for it to melt away.
This year the casualties in the spring mountains in Japan have been unusually high. Just two weeks ago four people died, one on Mt. Fuji, due to losing his footing on the ice and being unable to stop his fall.