Do you get to call yourself mommy, a woman who has had a child, if your pronouns do not reflect that state?

It's fine to confuse pronouns, if you wish. No one else has to use them, but you can ask to be called anything that suits you, on a personal level.

But a mother is a specific thing. It is defined as the female parent of a human child.

They/them here isn't a female, so should not consider herself by a female role or title, should they?