Yunessun can be found in Hakone and is one of the few places you can wear a swimsuit in an onsen (well, in the swimsuit section.)
There are different themed baths and outdoor slides in the baths too. There are also foot spas here.
Babies can enter as long as they are in swim nappies/diapers.
Small tattoos are allowed here if they can be covered (Yunessun can provide tattoo covers.).
It’s a good rainy day option with kids in Hakone (although I do also like staying at my ryokan when its raining and just relaxing and enjoy the onsen and food there too.)
See here for more details on the Yunessun website.
I have many additional suggestions for how to spend time in Hakone as a family here.