My DS is female. He was born female and utterly agrees that biology dictates his sex. We are 100% supportive and use his preferred pronouns and will support him whatever future holds.The same child was also a victim of abuse before transition.
I hope you do not encourage your child to take hormones or do any other irreversible changes. It is very understandable that they want to identify out of the female gender role, but your safest bet here is to fight for female spaces.
Without hormones and/or surgery, your child will be recognized as female and be able to use those spaces, if they still exist.
Since transactivists, the most vocal group, are not interested in separate spaces for transpeople, and even less in spaces where transpeople are segregated according to actual sex, the chance of getting that is very low.
I hope your child will be okay. 
I think it would be better to clear up gender vs sex. You cannot change your sex but you can change you your gender. Gender appears to be a personality trait from what I gather.
That is what the transwacktivists tell us.
But actually, gender is a tool by which men oppress women in patriarchy. That is what the word gender is used for in feminist literature.
Women cannot change their gender; males will molest and rape us regardless of how we identify. Not once in my life has a man asked my gender identity. Not before hitting on me, not before molesting me. Never.
Employers do not ask for women's gender identity before paying them less than men. They do not ask for gender identity when promoting men over more competent women.
Some female people might be able to pass for men and thereby gain some advantages. However, those advantages always come with the danger of being found out.
What women cannot do is just identify into male privilege. That's not possible. As long as men see that you are biologically female, they will treat you accordingly.
It is only men who can force women to treat them like women without plausibly disguising themselves as women. I wouldn't say that those men have changed gender, even. Gender is a hierarchy, with men on the top, and that is not changed by men invading women's spaces. Quite the opposite, actually.