If I were you, I'd activate the pass before going to Takayama. Going from Tokyo through there and Kanazawa, and then down to the Kansai area is close to ¥20,000 yen you could save. It would mean 2 more days in Tokyo at the end, but there is still lots to see in the area, and it would eliminate 2 days that you have trouble filling.
Switching Hiroshima and Osaka also makes sense, so you don't need to backtrack. About 2¾ days is way more than enough to see both Hiroshima and Miyajima, so you might add in Iwakuni with its iconic historical bridge+castle. Three days in Osaka is really two too many since you are already seeing Tokyo for an extended period. There are some places you can see in the day, but the city is far better at night and you can zip over there in the evening after seeing Kyoto when the temples all close down around 5 PM. Dotonbori, is one of the best places with some really good food, plus the night view from the Umeda Sky Bldg is spectacular. Instead, why not go see Kobe for a day? And since you have the time, go spend a day and a night at Koyasan.
For your other 3 undecided days, there are many possibilities - the half day hike up to [Konpirasan in Kotohira], (https://www.youtube.com/watch?v=ley9rVIewoQ "Konpirasan"), Kagawa, is wonderful. Going to Takamatsu is also nice - it has one of Japan's finest gardens, named Ritsurin.
Going deeper south is an interesting thought - few tourists go, so it is uncrowded, and there are some excellent places, such as Hagi which is a sleepy but historical area retaining its samurai neighbourhood, and Akiyoshido which is a 300 million year old limestone cave 1km deep with an underground river and giant caverns.
Kyushu is another great idea, and you can fly cheaply from Fukuoka back to Tokyo on a low cost carrier like Jet Star or Peach. The Nanzoin Temple is breathtaking and almost a classified secret from the outside world. There is also gardens, great shopping, parks, and its famous tonkotsu ramen. Nagasaki is another magnificent city to visit, with western and Chinese influences. And while few others mention this, you can easily and cheaply fly from Osaka or Fukuoka to spend some time in Okinawa, which is also unique and has some beautiful sights and scenery.
For the rest of your time in Tokyo, there are many good areas to see too. Look into getting a JR Kanto Area Pass and go see Nikko, Izu, Karuizawa, Kusatsu, or Kamakura/Enoshima for a day. You can also get around Tokyo cheaply with a 3 day subway pass.
Best of luck! I'm sure you'll fill up a lot of memory cards from your trip.