It was fun to find the best way to come to Kagoshima, my home town, from abroad.
I did not know there is so called "Japan Fly Pass".
If you take Finair, you must choose JAL because both Finair & JAL are member of oneworld.
https://www.jal.co.jp/yokosojapan/Even though airfare to Kagoshima is still higher than that of LCC, there are more flights. When think about flight delays or weight of luggage, I would choose Finair + JAL.
If you come to Kansai airport, just take a bus to Itami airport.
http://www.okkbus.co.jp/en/timetable/kix/t_itm.html#fareYou may be able to catch afternoon flight to Kagoshima. It takes 70 minutes.
You can also take Shinkansen from Shin-Osaka to Kagoshima Chuo by "JR Pass", it takes 4 hours +.
It seems like you concern about whether or not it is safe to drive there.
In that case, you come to Kagoshima Chuo Station by bus.
You can see how road and people look like then decide. If you think you want to drive, you may visit travel information center at the station and ask them to arrange. Toyota, Nissan, Mazda, Orix, JR, many choices.
They charge you per hour. If you need to come to Kaimondake
early morning, must rent within the day you arrive due to their office open hours.
If you stay at Centaral Kagoshima, perhaps you need to pay for parking. If you stay at Ibusuki, most likely parking is included.
Hope you find suitable way and enjoy staying there!