Looking for traditional ryokan on Izu Peninsula with Private Onsen for Anniversary

My wife and I will be in Tokyo next month for our one year anniversary. We really want to spend a night or two on the Izu Peninsula at a scenic ryokan preferably with private onsen options. (She has tattoos)
We’ll be relying on public transportation more than likely. What are some of your recommendations and experiences? TIA

by SuggestionSpirited68