As I mentioned, I also travel in mid to late June because of school, so I definitely understand.
Honestly, there is a lot to see everywhere in Japan. Kyoto has some really superb temples and shrines. I mostly go to Osaka to see a friend/shop and to visit Universal Studios. I honestly don't think Osaka offers as much to a tourist also going to Tokyo. But I think you need to honestly decide what your priorities are and what you would like to do. Since you're in the USA, when you say mountains are you thinking something like the Rockies or smaller and greener mountains like the Appalachians. Are you think coasts, like dramatic coasts or beaches or?
I don't know how old your child is, but are they old enough to have opinions on what they want to do? My first trip with my child I didn't give her a lot of input but on the most recent trips and our future June one, she will have a lot more say. We always try to mix places we've been before with new destinations. My child is in 3rd grade. Favorite places for her have been: Matsushima, Uji, Kyoto, Miyajima, Takamatsu, Tokyo (for shopping) and Nagano. Of the onsen she has gone to, her favorite was the one at the Associa in Takayama.
Kurama and there might be one in Arashiyama are the nearest onsen to Kyoto, though there are many sentos. Though Kyoto is full of traditional ryokans. Arima onsen is not so far from Kyoto.
If you're flying from LAX or SFO you do have the option to do an open jaw flight.
If you really wanted to do Kyoto too, I might do: 4 nights in Tokyo, 4 nights in Kyoto (you could always easily go to Osaka for dinner one night) , 2 nights in Kanazawa, 2 nights in Takayama, 1 night in Matsumoto and 2 nights in Tokyo, or something along those lines. I would probably not get a rail pass and just pay out of pocket, since the fastest way to Takayama from Kanazawa is bus and you would do a bus from Takayama to Matsumoto. I would prob skip getting off at Shirakawa Go, I think Hida Folk Village in Takayama itself is better presented. You could day trip to Shin-Hotaka or Okuhida. Depending on flight times, you could also add a night in Kamikochi or somewhere else on the way. This itinerary is more mountain heavy,
If you want to go north, I'd have to think more on it. We're actually currently looking at a Tohoku based itinerary ending in Kansai, though I would prefer to start in Kansai and end in Tohoku. We shall see.
Speaking as a parent from the USA, I don't know the age of your child, but my child always finds the time change to Japan challenging. She is up by 5 or 6 am the first week and ready for bed by 6. I find as an adult I adjust much better than she does. Please be cognizant of this when planning your itinerary.