Oof. I wonder if chartering a boat might be easier. I knew I should've gotten my boat license before I moved to the middle of nowhere, heh.
Some useful links digging a bit (google translate should mostly get you there if needed)
- http://www.town.minamitane.kagoshima.jp/sightseeing/H3f8.html
- http://www.town.minamitane.kagoshima.jp/sightseeing/rocketkengaku.html
- https://www.jaxa.jp/about/centers/tnsc/
The exclusion zone doesn't look too terribly big (though is cut off). I'll have to poke at that a bit more.