Kinosaki Onsen
All seven of Kinosaki Onsen's public baths welcome tattooed guests — any size of tattoo, no cover-up required.

With sources, not rumours
The towns that officially welcome ink — plus the private-bath stays that make the question moot.
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.
Many Japanese onsen still refuse entry to tattooed guests in their shared baths. There are two ways around that. The first is a private bath — what every property on this site offers. The second is to pick a town that has officially dropped the ban. Where we make that claim, we cite the official source so you can check it yourself.
All seven of Kinosaki Onsen's public baths welcome tattooed guests — any size of tattoo, no cover-up required.
Kusatsu Onsen's three public baths (Gozanoyu, Otakinoyu and Sainokawara) do not refuse entry to tattooed guests, according to the official Gozanoyu FAQ.