Looking for a nice affordable onsen hotel (~$100–150/night) for a 2-night stay in May. Any recommendations?

hey! i’m looking for a nice (but not super expensive) onsen hotel for a 2-night stay during a 2-week trip to japan in mid-may. i started doing research and stumbled on Kawazu / Nanadaru onsen (i see rooms for about €80/night right now) — it even has private baths, which sounds perfect.

i’d love recommendations for similar or even better places with private baths (in-room or reservable private baths) in a pleasant area where we could enjoy a traditional stay outside the big cities, but still easily reachable from tokyo / kyoto / osaka. our budget is about $100–150 / night max.
is mid-may a good time for an onsen stay — or is it getting too hot/humid so a hot spring would be less enjoyable? would 1–2 nights be enough, or even too long if it's warm?

any hotel/ryokan names, links, or area tips much appreciated — thanks!!