Both my parents were born and raised in Japan - however they moved overseas so my brother, sister and I were all born outside of Japan.
After each birth, my parents registered us as Japanese nationals to the Japanese government. I have dual-citizenship (two passports).
My siblings and I all have "Kanji" names.
Obviously the hospital records showing our names are in romaji, but the paperwork submitted to the Jp city-hall/government was all done in Japanese (kanji).
The important thing is registering the birth within the given timeframe (check with your local city-hall). As far as the Jp government is concerned, they will view your child as a Japanese national born abroad.
Hope that helps.