Sennennoyu Koman
Two free private baths in the annex, filled with source water (room baths are not onsen by local ordinance)

Officially tattoo-friendly
The public baths here welcome tattooed guests — source cited below.
All seven of Kinosaki Onsen's public baths welcome tattooed guests — any size of tattoo, no cover-up required.
Source: Visit Kinosaki — official tourism guide — https://visitkinosaki.com/plan/visitor-info/guides/tattoos-in-kinosaki-onsen/. Policies can change; if a shared bath matters to your trip, check the source before you go.
Prefer to skip shared baths entirely? Every property listed here has a private bath — a reservable private bath or an open-air bath in your own room — so tattoos are a non-issue, and so is privacy. These properties in Kinosaki Onsen all have one:
Two free private baths in the annex, filled with source water (room baths are not onsen by local ordinance)