Unless the money issue is getting really tight for you, I suggest you get the 14-day pass. Also as mentioned, the heat and humidity will be getting pretty thick by that point, and the rainy season about to seriously kick in. Expect the weather is abysmal at least 1/3 of the time, and have a backup plan (museums and aquariums for example).
The good news is hydrangea season is really nice, and there are many great temples in Kamakura to enjoy them (Meigetsuin for starters, but also Hasedera and others). In Kyoto, the Yoshiminedera Temple is another gem. I also think taking a day off of Tokyo (which would leave you with 3) and going to see Nikko is a great idea and a good escape from the heat. If you want both Nikko and Kamakura Enoshima, you could pull a day off of Kyoto - seeing Nara Park is totally worth it but you can do it in half a day.https://goo.gl/LfvWnL
You'd still have time for Osaka in the evenings after seeing Kyoto - evenings are best there anyway, for Dotonbori, Shinsekai and many great city views.http://goo.gl/XE5R94
On another evening, you can also go see Kobe.
Two days are more than enough for Hiroshima & Miyajima - so if you like, along the way there or back you could see Himeji, with the country's finest original surviving castle, and the Kokoen Garden nearby is also excellent.https://goo.gl/65ZtRi
For Hiroshima itself, so many just rush to the Peace Park and leave, but don't cheat yourself like they do - the Park is good and sobering, but there is a lot more, and not overrun by the tourist mobs, like the Mitakidera Temple and Shukkeien Garden.https://goo.gl/fAz4ax
If you are staying overnight on Miyajima, be sure to see the Itsukushima Shrine at both high & low tides. http://www.tides4fishing.com/jp/hiroshima/itsukushima
At low tide, you can go out on the beach to the giant torii gate. And if you can get there in late May before they cover up the torii gate for 12+ months of renovations in June, you are one lucky person.https://www.japan-guide.com/e/e3401.html
For Koyasan, if you are doing the temple stay, you might bring a few snacks with you - some people have said they felt hungry later since the Buddhist vegetarian food is so light and doesn't leave you full for long. There is a convenience store there but you might have to walk 40 minutes round trip to get something. If you want to see the Okunoin cemetery at night (a surreal experience) be sure to pick a temple with no 9 PM curfew. You don't need any tour - you can just go yourself.https://awesome-tours.jp/en/tour/night-tour-return-bus/
Your biggest question is about Hakone - and in the end, only you in hindsight can really say if it is worth it. Yes, it is touristy - flooded with them in fact. If you go really early and do the Loop differently (clockwise) than the crowds, you might be able to escape some of the chaos. Consider a good view of Fuji as a big bonus, but don't get your hopes up - it frequently is bad. You can always look at the live web cam and check the weather report to increase your chances, if you are flexible.http://www.hakone.or.jp/en/web-camera/https://is.gd/f1gOsc
The best views of Fuji are usually early in the morning or evening.
Aside from the JR Rail Pass, there are other passes for Hakone, Nikko, Koyasan etc you should look at.https://www.japan-guide.com/e/e2357.html