Long time lurker, first time poster. As such, forgive a cautious disclaimer from a newbie taking a tentative step into unfamiliar territory. I respect others' right to claim a gender identity - I've met transpeople to whom this is quite clearly innate - and acknowledge the possibility that, over time, science may help us to understand this more fully. I'm also appalled by the impact some of the more militant TRA organisations and individuals are having on freedom of speech and women's rights, and am, perhaps, even more disturbed by the lack of critical thought and naive virtue-signalling that sometimes seem to drive this. I don't know if this makes me gender critical or not (I've been wanting to ask!) I'm certainly critical in the sense of taking an academic interest in a hugely complex issue, and I'm critical OF (and increasingly disturbed by) some of what I see happening in relation to it.
Anyway, I'm also a teacher and wanted to put in a word from that perspective. Gender ideology arrived at my school relatively recently, with (so far, quite mild) exhortations to use pronouns and colourful Genderbread powerpoints. We also have some socially transitioning teens. And some of the staff, at least, are finding that navigating this, as professionals with a deep sense of responsibility for the wellbeing of all the children in our care, is exceptionally difficult.
In this context, I think the best starting point is to recognise facts and acknowledge nuance. Both sometimes seem to be side-lined because of their power to problematise any reductive hardline. So, re: the lesson mentioned, I'd start by acknowledging that something may have been lost in translation between the classroom and home and suggest investigating further before taking more decisive action - starting with what's certain. The raw facts seem to be that 1) the daughter came home convinced of the teacher's having taken a "side" & 2) the vote was hands-up. In this context, the best-case scenario does seem to be that it was a badly-handled "debate" in which the teacher unintentionally let her opinions show, making it difficult to impossible for students to formulate and share their own. That's problematic in its own way, but at least it's not cynically deliberate. The worst-case scenario is, yes, that it was the kind of indoctrination suggested above. Frankly, either, in my experience, is eminently possible!
Last but not least, here's another more complex possibility that I can also see, although it does seem very unlikely. The teacher may, personally, have an open-minded respect for JKR... but not have felt able to say so, and, instead, mutilated her explanations in an anxious effort not to offend. I'm not saying I'd excuse this for a second, and it's not something I'd do myself. I've challenged inaccurate and troubling - fundamentally homophobic - material in this area in my own school. But... it really wasn't easy to do this, and careful chats with management are the least of it! It's extraordinarily challenging to navigate these issues under pressure in front of a lively class: to speak spontaneously, manage interruptions and low-level behaviour, monitor 30 students for confusion and insecurities... and all the while to find a way, in my choice of words, to respect all interests equally, whether gay, trans or feminist; to acknowledge the class' precocious "knowledge" of it all, while nurturing a deeper understanding; to juggle conflicting opinions and the bubbling hormones fuelling these... Honestly, each time, I feel a little like Winston leaving Parson's apartment (exit, pursued by a slingshot) because of the sheer volatility surrounding these issues. It's really, REALLY hard. In wanting to advocate, above all, for tolerance, the right to be who you are and freedom of debate, you feel, yourself, horribly constrained. This is why I'd investigate more thoroughly and tread more lightly first (who provided the lesson plan, how long did the teacher have to prepare before delivering it, what's the school's own stance on all this etc.) Teach the school that fact, nuance and empathy matter.
And then, yes, after that, OfStEd & various other suggestions above if the answers justify it, because freedom of speech and thought in our schools matter. So, SO much!