IMO in young MC girls this view is usually based on a perception of oppression of transpeople, based in a kind of "be kind to the oppressed" conditioning of UK MC education and parenting.
If I were going to work on trying to get her thinking - which I would only do on a much separate occasion - I would work to undermine this argument.
So, middle class female from stable family sharing bathroom with transgirl - no clash of rights perceived. The female would not feel threatened and would perceive she is being inclusive to a less fortunate or less privileged "class" - transpeople.
However, flip that about a bit. Now, it's a Muslim woman in hijab in the bathroom when the trans woman comes in. Both religion and gender reassignment are protected. Religious Muslim women should not be undressed in a space with a male person who is not their husband. Who is more oppressed in this scenario? The religious minority or the LGBT minority? Would she recognise the internal conflict the Muslim woman might face in this bathroom scenario? Is it simply because she is bigoted? Should the Muslim woman stop using public bathrooms so she doesn't face this scenario?
Now it's a woman who was recently raped on a night out and who has PTSD. Her friends have persuaded her to come out for a meal and she has agreed after much anxiety. As she exits the cubicle, she sees a trans woman come into the bathroom. She feels a wave of fear as she can tell this is a natal male, and she is alone in a vulnerable space with him. Now who is more oppressed? The sexual assault victim having a physiological fear response, or the trans woman? Is the rape victim only feeling fear because she is a bigot? Should she not have come out at all if she didn't want to find herself in a secluded space with a natal male person?
Now it's a little girl, aged 7, who has popped into the ladies' loo at M and S while her Mum changes her little brother's nappy in the separate baby changing room. A trans woman emerges from a stall as she comes in. The 7 year old feels confused and scared. She runs to a cubicle and locks herself in, then starts to cry. Age is a protected characteristic alongside gender reassignment. Is she only crying because she is a bigot? Should she not have been allowed to use the ladies' loo alone unless she had been prepared that sometimes she might see someone she perceives as a male person in there, and that's ok?
I think sometimes people with great privilege don't consider that their experience is not the universal one; and that because they would not feel threatened by a transperson in a particular scenario, this doesn't mean that nobody would. That it isn't the case that all women as a class hold more privilege than all transwomen as a class. And if someone does feel threatened, is that only because they are by definition transphobic or bigoted, or is it possible that there could be a valid reason? If there is ever a valid reason, then there is a clash of rights, and a discussion should be had to work out a solution.