Rock onsens Ryokan in Hakone or Kawaguchiko

I'm looking for a ryokan in Hakone or Kawaguchiko, that has outdoor/open air rock baths

Budget for 2 people is around 500$. Additionally, I'm interested in being able to reserve an onsen for private use, kaiseki experience (even better if food is served in the room), and generally a nice surroundings (garden, nature). I've already done an extensive research can't seem to find anything, most of the ryokans seem to have "normal" bathtubes filled with water.

Thank you in advance

example 1 https://www.google.com/url?sa=i&url=https%3A%2F%2Fthegate12.com%2Farti...

example 2 https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.alexrockinjapan....

Hakone Kowakudani Onsen Mizunoto has similar conditions, but it is reasonably priced and may not be quiet as many children visit.
It's not a private bath, so it's difficult to take pictures.
https://www.hotespa.net/hotels/mizunoto/
https://www.ikyu.com/caz/00030982/

Bansuiro Fukuzumi might also fit the criteria. This is also not a private bath, so you won't be able to take pictures.
https://www.2923.co.jp/

I think Hakone no Mori Okada is also within budget.
https://www.hakonenomori-okada.jp/