I'm a biology teacher and have had this question before
In the genetics topic we actually talk about certain genetic diseases on the GCSE spec (i share my cousin's story of living with CF) and part of that is talking about how we cant change the genetic code in all of our cells, this is why CF treatment is so fraught with obstacles.
So pupils are already aware of how the DNA in every cell can't be changed in a whole organism, even with gene therapy (which is also on the spec)
Then it becomes a distinction between sex and gender, and every school in the UK is bound by the equality act 2010 which gives both sex and gender reassignment as protected characteristics. So the answer then (legally) is that people cant change their chromosomes but people can change their gender.
Its also a good way to show the very rare cases where chromosomes don't always perfectly align with sex (like Klinefelters, Androgen insensitivity, Turners, etc) Pupils tend to enjoy knowing when the GCSE curriculum is "lying" to them, lol. It happens a lot in chemistry but there are also bits of bio where they like to know that they aren't learning the whole story - they always ask about green eyes for example, when the GCSE only ever talks about blue and brown!
Ive never found this to be confusing for pupils, its a pretty clear distinction.