Justifying purchasing JR Pass

Hi,
Unfortunately, a 14 day pass would not pay off for your itinerary above. But you don't need one anyway. You could get a 7 day pass and use it July 27-Aug 2, and you would almost break even. You can use a regional pass to see Fuji, like the Hakone Free Pass, and afterwards go to Mishima where you can catch the bullet train to Kyoto. If you take half or a full day to see Himeji or some other not-too-distant place near Kyoto, you can start to reap some good savings from the pass, and Himeji has Japan's finest castle, newly renovated.

Best of luck.

Of course, if you would like to do more traveling and expand your itinerary, it is not hard to make a 14 day pass cost effective, and there is no shortage of great places to visit.
Okayama is not far and it is famous for its castle, plus Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
Hiroshima also has a number of good places. Miyajima is the most famous, and aside from Hiroshima itself, Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain. Taikodani is also a beautiful shrine in Tsuwano, Shimane. Hagi Castle was actually delightful, and the preserved samurai district and Mori related temples a rarity. One of the very best is the Akiyoshi Limestone Cave, a 300 million year old wonder with huge caverns and underground river running through it.

If you can, try to find a little time to see the Japanese themselves. Go visit a Japanese supermarket and see what people typically eat. Look through a park and watch the Japanese kids play. Learn about the history of some of the places you visit before your departure - it will give your trip so much more meaning.

All in all, a lot more to explore and enjoy.