Hi!
I totally get needing to go when you need to go. We go mid-late June until early July because that works with my daughter's school schedule. Lots of people will say it's hot and humid and rainy and miserable, but I actually don't mind too much. I think the worst part is the airfare tends to be high then. Honestly, for your time period I think the worst thing is it's break time in Japan so accommodations might be more expensive, but the Pound is stronger than the Dollar which is stronger than the Yen, so it is what it is.
Kanazawa will still be hot, but I think it will be less crowded. Kanazawa certainly felt less crowded than Namba last June. I seem to remember Kanazawa feeling a little cooler in the evening once the sun went down more so than Kyoto or Tokyo or Osaka. I imagine Tojimbo will still be hot as well. I went in November to Fukui a really long time ago. I honestly remember the squid ink ice cream more than Tojimbo itself. If you eat ice cream, Kanazawa will have a lot, some quite photogenic with the gold and platinum leaf. I've not done the Alpine Route... It doesn't really appeal to me. However I went to Kamikochi in late June last year. It rained and I was freezing. We were in long sleeves and a jacket and leggings. I don't know what temperature it was, but it was cold! I do think the rain made it worse. I suspect you will need a light jacket for the Alpine Route, even in Summer. This is the official webpage:
https://www.alpen-route.com/en/I personally find 20 C when you are use to 32 C feels very chilly. While right now 20 C feels positively balmy since it's been 10-13 recently. It sort of depends on what temperature you're coming from.
I found Nikko sweltering still when I went in August 2000. Kurama near Kyoto could be cooler? I went in April. I went to Enryakuji in late June in 2010 but I don't remember how hot it was. It's also possible Ohara is a little cooler? I've not been though we might go in June.
For accommodations I don't know your price range but will mention a few places and things I've noticed. I find price wise the mid range hotels cost more in Osaka than in Kyoto, Kyoto has a lot more well priced 3 star hotels with slightly larger rooms than Osaka. If you're looking for hostels, I think prices are better in Osaka. In Kyoto, I loved the Hana Touro Hotel near Yasaka jinja. It was right near a Keihan and a Hankyu station. I also like the Royal Park Sanjo. It was quite convenient. I'm trying the Vischio this time and there is a new Mitsui opening near Kyoto Station. I've not stayed there, but I do like the Mitsui chain after trying to the Otemachi branch in Tokyo, which I would recommend. I won't use an APA.
If you have the time in Kansai, you could go to Wakayama-ken. There are beaches in Shirahama and Tanabe that are quite pleasant. The mountains could be cooler, the same with parts of Nara-ken. It rained when I was in Nara last so it was quite chilly. We're trying the Sakurai area this trip and I can let you know if it is very hot when we are there. There is a direct airport Limo bus from Wakayama station to KIX or it's a not so bad trip by train. Wakayama is actually a little closer to KIX than even Osaka. There are lots of onsen in the Kii mountains.
I would make sure all hotel rooms you book are an actual double or twin. My daughter is younger than yours, but I still would not want to share a single with her, it'd be awful tight, both in the room and in the bed!
Good luck!