Sorry I'm come to this thread a bit late but had thought about the issue for a while.
My main argument to your daughter from my own experience is that the secret of happiness, or at least contentment, is accepting your reality. If you are female then adopting a male persona, taking hormones, God forbid binders or surgery, will never make you happy. Your daughter will only find peace by understanding and living her truth, whether she's a lesbian or gender nonconforming. She's female, with a female body. Being trans is an ephemeral trend. We recently had a thread where there were 5 transmen in one school year, ffs. Ten years ago this phenomena was unknown. Girls transitioning is a terrifying fashion, but it is a fashion. We just have to hope we can steer girls away from making long-term changes to their bodies. It's a hell of a lot more dangerous than becoming a punk or a Goth.
It's very noticeable that the only transwomen who seem at ease with themselves are the ones who are happy to acknowledge that they are male and that this can never be changed.
I had lots of body issues when young, though never wanted to be a man. It took time and effort to get to a place where I was comfortable in my own skin. But that is the secret. Loving your own flawed female body, and owning it.
It's not that being a woman is all that different from being a man. We're all people first. Being a woman isn't very important to me, I'm Prawn, and I am a woman. It's a fact not an identity. But - and this is the important bit - I never feel the need to lie about or reject who I am. Being transgender is all about pretending to be someone you are not. It's the opposite of being at ease with who you are, and as such can never result in security or long-term happiness.