No, because men still have an automatic advantage, even in your organization.
If you hired a man for that position, they immediately gain the privilege of using the men’s bathroom, a luxury barred in your organization for women you hire.
Equally qualified. One gets the nice bathroom. The other does not.
You set up your organization so that the men’s room is a privilege that you want to deny employees. The only way to do that is to hire women.
Systemic discrimination. Your organization is the system of oppression. Congratulations, you’ve done it.