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I'm gonna answer your questions first and then give you my 2 cents about something you absolutely did not ask about lol
Regarding going to Karuizawa on the way back from Kansai to Tokyo: It's kind of a detour, in that it would be faster for you to go to Tokyo and then Karuizawa than to go to Karuizawa by going through Kanazawa and all which is what I assume you're going to do. If you want to do that route it would be 5 hours just to be in Karuizawa for very little. If you want to see Karuizawa, I would do it on the way to Kanazawa on the 3rd. That would save you a lot of time. Instead, you could go directly to Hakone and stop somewhere along the way, like Nagoya or Hikone, Hanamatsu or Atami if you want a break from travelling or to grab lunch or something.
For being able to see Mt Fuji from a distance honestly it's luck of the draw. Yes July isn't the best month but I've seen very clearly from much further away in July quite a few times. What I'd say is all those areas (Kawaguchi, Hakone, etc) are quite close to each other and you could book hotels that are cancellable and adjust based on the weather forecast. Last minute could be more expensive (but not always, I've booked a 28$ night last minute because I missed the last train back to Tokyo, it's feasible!) but if this is really important to you, then you can do that.
Now
I know it isn't an answer to your question but I will just say, I would NOT climb Mt Fuji on the second day while you're still adapting from jetlag, unless you're coming in from a nearby country where the jetlag won't be as much.
I climbed Mt Fuji in 2018. I had prepared, I had hiked up many mountains in the weeks preceding, increasing height gradually, etc, sometimes doing 4 peaks in a day to be extra ready, and I thought I would be okay. People said it was an easy hike.
It was not an easy hike, at least not to me and not compared to the other ones I had done (many of which were rated as more difficult than Fuji by the guides I used).
I got my ass HANDED to me by this mountain, and it was very humbling to see elderly people struggling much less. One of the main things that made it difficult was adapting to the altitude. I got knocked off my feet by the wind twice. I did not realize that some places would require me to use my hands to scramble (so had not brought gloves, that one was on me).
You say it's a guided tour and your itinerary implies it is on two days but I'm not exactly sure where they are having you spend the night so here is my advice:
(If you are used to high altitudes and live like, idk, in Colorado, disregard):
If I had to redo it (glad I did it but NEVER AGAIN) I would sleep at 5th base the previous night to acclimate to the altitude at the very least because I felt out of breath just walking horizontally. Honestly, even doing it up and down in the same day is something a lot of people don't advise, though personally, I think that part is fine if you start really early and you're acclimated.
I know a lot of guided tours make you climb most of it, sleep at the second to last station and then climb the rest in the darkness to go see the sunrise at the top. I think sleeping at 5th station the day before is a much better idea. Just my 2 cents.
I would say it took me a solid 6-8 hours to acclimate, of course, it varies by people. The mere idea of doing this while jetlagged? I really, really don't advise that. Especially considering you are with a teen that is likely still growing and all. I'll add that I don't think you necessarily need a guide or group for Fuji. Very easy to figure out on your own and allows you to set your own pace which personally was a godsend. When I climbed, very few groups I saw actually made it all the way to the top. Most people were giving up before the end.
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