Born in Japan. No Japanese passport, not a citizen, not a resident. Need birth certificate. Am I screwed?

I'm applying for a British passport as a current citizen and resident of New Zealand (for dual citizenship). Both my parents are foreigners / non-Japanese. I was born in Tokyo (but my dad was born in England so I'm valid). The issue is I need my original birth certificate (in Japanese) and I only have copies. I've visited my local Japanese Consulate, and they've basically told me to try emailing people online.

I've sent an online inquiry to the ward I was registered in, but I feel a bit hopeless. I've spend so much money on this damn passport application (thinking my 'certificate of birth' english document was enough – but it's not since its a 'hospital record of birth' not an actual birth certificate), and every time I send a document over (minus $60 each time), they wait a month before telling me it's not valid. My application expires June 18th, is there anything I can do here?

Has anyone been through something similar?