Kyushu Kirishima, Kagoshima
Bihada no Yu Koshikano Onsen
美肌の湯 こしかの温泉
Name in Japanese. Look for this on the sign, or show it to a taxi driver.
Private bath onlyStated by the facility
Hot-spring bath. Hot spring (spring details not checked)

Tattoo rule
Private bath onlyStated by the facility
What the official page says
タトゥーがありますが入浴可能でしょうか? 可能です。当施設は全室、浴室が付いたお部屋となっており他のお客様の目を気にせずご入浴をお楽しみいただけます。
In short: Every room has its own bath, so guests with tattoos can bathe without sharing.
Where it is
Kyushu region, Kagoshima prefecture. About 203 km in a straight line from Fukuoka.
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Rules can change. Check the official page on the day you go. Any paid link is labelled next to the button. We have not visited this bath ourselves.