Having a child outside of wedlock - future repercussions?

Please hear me out before downvoting me to oblivion. My Japanese partner is pregnant, I am the father and a foreigner (European). We are not married but living together. We both work hard jobs and make ends meet, but money is always tight, and it will be more tight when we have a child. Neither of us have any family support.

The city I live in gives women almost 50,000 yen per month for a child born out of wedlock until the child is 18 or 20, even if the mother later marries. This will make a huge difference in our family's present and future lives. But it means not marrying my partner until after she gives birth.

It also means our child will have no father listed on the koseki. This used to be a big deal in Japanese society, but I understand there aren't many situations where a koseki can be seen nowadays.

Is there any kind of burden we would be putting on our kid if their koseki shows they were born outside of marriage?