Yuyado Tokino Niwa
Rooms with open-air bath

Officially tattoo-friendly
The public baths here welcome tattooed guests — source cited below.
Kusatsu Onsen's three public baths (Gozanoyu, Otakinoyu and Sainokawara) do not refuse entry to tattooed guests, according to the official Gozanoyu FAQ.
Source: Kusatsu Onsen — Gozanoyu official FAQ — https://onsen-kusatsu.com/gozanoyu/faq/. 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 Kusatsu Onsen all have one:
Rooms with open-air bath
Half open-air bath rooms; hourly fee-based private outdoor bath
Every unit has a private hot spring bath (open-air not specified)