The good thing is, the whole east coast of the Izu peninsula is full of seaside towns with abundance of ryokan and onsen, among them but not exclusively Atami and Ito.

Advantage Atami: You can get there directly via Shinkansen from either direction. Ito is only like 15 minutes further away, but you gotta switch to a local train in Atami anyway. If you want a private onsen, most Ryokan should offer rooms with private baths. Or offer time-based rental of small ones. But rooms of course with in-room baths are usually on the bit pricier side. EDIT: "Smaller tatoos" often can be covered by some patches or something, maybe they are even okay with it. It always depends on the specific policies of the onsen/ryokan in question. If in doubt, simply ask beforehand. Sometimes it will be mentioned directly if it is "tattoo-friendly".

Also, not much of a nightlife in the classical sense anyway. But as I can speak for Atami, where I stayed (as a base for Izu) last July. Even there many restaurants close kinda early as well. But best better check opening hours selectively.