Adding my birth to my mother’s koseki

My mother moved to the US around 50 years ago when she was only 5 years old. She doesn’t speak any Japanese and didn’t even realize that she had Japanese citizenship until I found my grandmother’s koseki in the attic. I was able to get a new copy and send it to her so she can file for a Japanese passport.

In the meantime, I want to add myself to her koseki. I know that I’m too late to be added as a “full entry”, but I was wondering if there was a way to record my birth under my mother’s entry, similar to how a non-Japanese spouse is recorded. This would make it easier for me to get another copy later and to prove that I’m the child of a Japanese national.

I tried contacting the city hall by phone to ask about this, but they seemed confused. Perhaps my Japanese wasn’t good enough to accurately ask about this. I gave up rather quickly because I became unsure whether what I was asking was actually possible. Does anyone know if this is possible?