There's no difference between someone who says they are the opposite sex for one reason and someone who says they are the opposite sex for another reason.
Whether you are doing it to prove a point or not, you are still not the opposite sex and your presence in a space for the opposite sex is still going to be at best an irritation and at worst a danger for the people that space is actually for.
Think about it for a second.
Women calling themselves men in order to access the men's single sex pond do not represent any danger to those men. They aren't going to set off any trauma responses in men who have been raped or sexually assaulted by a woman. Probably the worst they can do is make the space unsuitable for a man whose religious beliefs prevent him from sharing that space with women. And yet this is unacceptable.
Men calling themselves women for any reason do represent a danger to the women that space is for. Statistically, the risk that they may actually harm those women is significant enough that it should not be ignored. If they do actually try to harm a woman in that space, for example, by touching her in a way that she does not consent to, she probably cannot defend herself because he is strong enough to physically overpower her, whereas a woman in a men's space is not strong enough to physically overpower the men. Statistically, a large minority of the women in that space will have been raped or sexually assaulted by a man, and may find his presence traumatising for that reason. We also know that a greater number of women are restricted from using mixed sex spaces like this due to their religious beliefs. For example, Muslim women have much greater restrictions on exposing parts of their bodies to strangers than Muslim men do.
And yet, the men are allowed to object to this, even though the impact on them is nowhere near as great, but the women are not.
It's misogyny on steroids.
I don't give a fuck why a man is saying he's a woman, whether he believes it or not, or how sad he will be if I don't go along with it. He's not my problem. He shouldn't be any woman's problem.
Aunt Lydia is exactly the right term for the women allowing this.