Good luck.
My husband is Japanese and I spent several years living there, teaching in an English immersion school for Japanese kids (I taught 2 year olds.). Giftedness doesn't seem to be well understood or acknowledged in Japan, as my husband doesn't seem to get our kids. He understands "high IQ", but doesn't get how that doesn't necessarily equal "good grades".
As you're probably aware, Japan is very test-focused and success in school/life seems to be solely based on how well you can do on standardized tests. It's really unfair. Putting our son into a Japanese school would likely be a horrible experience for him, but he's not a schooly kid (We don't have him in an American school now. He's 6 next week.)
Homeschooling is not common -- and maybe not even allowed -- as far as Japanese citizens are concerned. However, I doubt that anyone would make a fuss about what you do with your non-Japanese kid. That's the somewhat good part about Japan. As a foreigner, you get to follow your own rules. I have a friend who homeschools while living on the American base in Okinawa. She loves it. I could put you in touch with her if you'd like.