Japanese cities are typically more expensive than comparable American cities. Tokyo, for example, was once the most expensive city in the world. I think it's since lost that dubious title, but Japan is still expensive.
I've spend lots of time in Japan (retired Navy) and never once did I hear anyone mention buying a ring in Japan because it was a good deal. Hong Kong, on the other hand, is a great place to buy engagement rings. But then you're not going there.
Anyway, knowing what I know of Japan (and Hawaii for that matter), I think you're better off buying in Hawaii.