Family register [koseki] and parental right or duty are not perfectly linked.
Your girl's family register is not to show who has her parental rights and duties.
Your girl was born in Australia to her Australian mother
and therefore did acquire by birth the Australian nationality.
In that case, as AK says, her Japanese nationality needs to be reserved.
At "日本国籍を留保する" in the birth notification form,
you the notifier sign and put a personal seal.
Japan adopts the policy of single nationality.
On the supposition that the rules on the Japanese nationality remain the same,
she is demanded to choose one nationality by the time she reaches the age 22.
This time, a new family register is created for her,
because her parents were not legally married at the time of her birth
and her mother is not a Japanese national.
When your marriage to your partner is notified,
a new family register is created for you.
Then, due to your affiliation,
you can make her enter your family register.
Do you know what municipally is managing your family register?
(The area of this municipally is called 本籍地 .)
Consulate-General of Japan, Melbourne
- 出生届 (birth notification) https://www.melbourne.au.emb-japan.go.jp/consular/reception/familyandn...
Ministry of Foreign Affairs of Japan
- 入籍届の書式 https://www.mofa.go.jp/mofaj/toko/todoke/koseki/pdfs/28nyuuseki.pdf
Following the succession to the Imperial throne,
the era name 平成 was changed to 令和 [ れいわ ] from May 1.
Now it is 令和元年 in Japan.
So, 平成 used for the notification date has to be corrected. I have actually completed the 胎児認知 at the consulate here
You have legally affiliated the girl before birth (胎児認知する) to yourself.
At the time of her birth,
your affiliation took effect, therefore you became her legal father.
That means, you already have parental rights and duties as her parent,
regardless of whether her birth has been legally notified
or how she appears in her family register.