OP, there are lots and lots of opinions on this thread about what being non binary means and most people seem to be of the view that it is all a load of nonsense.
However, to your staff member, it is not a load of nonsense.
The difficulty for you is that we are living in a political climate where the prevailing narrative is that people's gender identities must be "affirmed" and that any failure to do so is "transphobic".
I could write an essay about the hypocrisy of people who insist on being referred to in exactly the way they wish but call other people "cis" with impunity, or who voluntarily exclude themselves from the word "woman" but then want us to come up with euphemisms for female people such as "people with cervixes" and use them so that they can be included. I could speak at length about people who insist that the trans community is the most vulnerable and marginalised and their lives are in constant danger, but think it's fine for male bodied sex offenders to be housed in women's prisons and that if trans activists stalk JK Rowling outside her house and post her address on Twitter she's just getting what she deserves.
But none of that is relevant to your predicament.
Four things so far stand out about your staff member's behaviour:
- undermining their colleague by attempting to correct their language in front of children
- saying they feel "triggered" by middle aged women in senior positions, which is both sexist and ageist and also suggests that they are unlikely to be cooperative with anyone in that position who has management responsibilities for them
- making another staff member feel uncomfortable by commenting on her Harry Potter poster
- constantly correcting the children which they find upsetting and which detracts from the purpose of their interaction, which is for "them" to teach the children the curriculum in a supportive environment
This does need to be raised with "them", in the same way that it would need to be raised with any other member of staff without a non binary gender identity who behaved in an equivalent way.
However, "they" are unlikely to respond well to such criticism and are likely to make accusations of "transphobia".
So what do you do?
WRITE EVERYTHING DOWN. Cover your back.
Keep your own notes with dates and times, what happened, who said what, what the outcome was.
At the same time, every single time there is an incident, compose an email, very carefully, setting out exactly what happened, who "they" were interacting with, who said what, in what context, why "they" were upset, what "they" did about it, what impact this had on the other people concerned. Try to get absolutely all the relevant information down, commenting as little as possible on "their" gender identity, and expressing no opinions about that aspect of things whatsoever. Stick purely to the facts.
And then send the email to someone senior to you who has a legitimate need to know these things, for example, your own line manager.
Frame it either as a "this happened, just keeping you in the loop", or even better, if you can, "This happened, what should I do about it?"
Or, "This happened, what should I do if it happens again?"
Or "This happened, are you happy with the way I dealt with it?"
If this person has poor performance in any other aspect of their job which is unrelated to their gender identity, note that too (provided that it is something you would make a note of if any other member of your team did it - you don't want this to look like a vendetta).
The reason I say all of this is because "they" appear to be a prime candidate for being let go because they are useless at their job, and then bringing an unfair dismissal claim on grounds of transphobia.
You need to make sure that your own conduct is irreproachable.
But at the same time, you also need to do your job as a manager and if this person's gender identity is affecting their job (which it is), you can't just pretend it's not happening.