Unless you're going to Yunessun in Hakone, I would do neither. I really like Kamakura but if your older son is really an "average" US teen he's going to be over temples and shrines by that point, so Kamakura is not all that appealing. I personally find Hakone dull. Unless your boys are really into small gardens, small museums or taking lots and lots of different forms of transit, that is what you will be doing. The big thing in Hakone is the transit circle route involving train, funicular, ropeway, boat (my least favorite part) and then bus.
Now, if your family loves that sort of transport thing, then do Hakone. If you are just going because you want to onsen or go to Yunessun, you can day trip! A friend who was too shy to visit a non-bathing suit onsen with me went to Yunessun with me from Tokyo. After spending all the day time hours there she went back to Tokyo and I continued on to Kansai. You can substitute another onsen for Yunessun, it doesn't matter. Hakone takes a day when you want to do the transport loop, particular with the small museums and gardens. Just going to Hakone from Tokyo is only about 2 hours. But honestly if you're just going to use an onsen, I would try Kusatsu instead, the onsen are nicer.
BTW many people do Kamakura as a day trip from Tokyo. It's only 1 hour from Tokyo. One thing you can do is split up and part of the family stay in Tokyo while another part goes to Kamakura. So say you and your father could see Kamakura while your husband takes the boys to the new VR ninja experience.
You can not drive into Kamikochi. It's car free. You will need to drive to the Hirayu Parking Area, park the car and then pay for the shuttle bus to Kamikochi. http://www.kamikochi.org/plan/access#car
So you will be paying: Rental car, gas, Hirayu parking fee, and plus an additional 2050 fee per adult round trip to Hirayu parking and 1030 for your younger son. (see here: https://www.nouhibus.co.jp/english/routebus/kamikochi_line.pdf
It's 5040 per adult to take the bus from Takayama to Kamikochi. There might be a pass that includes more things.
If you have the time, you can certainly then take your car and continue onwards to Shirahone. https://www.japan-guide.com/e/e6042.html
Just be aware that day guest hours to use onsen can be limited and make sure to give yourself plenty of time and gas to make it back to Takayama. You could also just visit all the many beautiful onsen in the Okuhida area or take the Shin Hokata Ropeway or heck, both @LikeBike and I loved the Hirayu no Mori onsen near the parking lot.
Looking at your new itinerary, I would think a rail pass is not going to pay off. As you have in terms of long distance train travel: Osaka to Kanazawa 7130 yen (non-reserved seat, since this is a starting station getting a seat is not that hard. I've done it.)
Matsumoto to Nagano 3000 (reserved, the price is cheaper unreserved)
Nagano to Tokyo 8400 yen (reserved so you can take a Kagayaki or Hakutaka)
Tokyo to Kyoto round trip 27600. (roundtrip)
Tokyo to Narita (3000 yen)
However, you might benefit from the following:https://www.japanican.com/en/special/tours/shinkansen/index.aspx
Which would allow you to take the much more frequent Nozomi. The seats are non-reserved, but from Tokyo to Kyoto that won't be an issue, and I would think you can manage it from Kyoto to Tokyo since you will have so many more train options as well.
Now whether to get a pass or not goes out the window if you have expensive day trips planned that are not mentioned in the above itinerary (like Matsushima from Tokyo or Hiroshima from Osaka.)
Good luck and I hope you have booked lodging for the Gion Matsuri already.