OP I really feel for you.
It's plainly not realistic to expect the girls to make any waves themselves, and it shouldn't be down to them. I agree with pp that there's something bullying-ish about how the child is behaving. We all (parents, schools, society) need to give girls the belief that their boundaries are legit.
One option at this stage is just to try and get them to justify the decision. You could just ask a series of questions.
Is there a written policy on this & if so what?
If it's not a written policy who took the decision that pupils will be allowed to use changing rooms based on self identity not sex?
As pp' have said - did they do an equality impact assessment or a risk assessment?
How exactly is it justified by the EA. It might be legal, but it has to be proportionate & legitimate. How have they evidenced that? Why didn't they choose alternative, also legal options? You can use the EHRC guidence to show it's legal to provide separately for this child.
How do they know that their decision/policy is not discriminating against the girls on the grounds of sex, (or religion/belief?)
Copy in the governors or write to them separately. Do they know?
I really feel that if they have to sit down and work out the justification, it will crumble.
Do you know if the child plays sport with the girls when they separate by sex?
Separately, you could write to your mp and ask them to raise it with the minister for education, or get them to forward your concerns to the minister. You could ask what the new guidence they are working on will say on this, & state what you think it should say! They need to hear from real people in these real circumstances.