When JK Rowling says her "biological sex" is a key aspect of her identity, it's nonsense for a few reasons.

Firstly, nobody is gonna change your biology, so what exactly is it you are afraid of changing here?

Secondly, if you understood what it meant for something like that to be a key aspect of your identity, you wouldn't feel so glib about denying key aspects of other people's identity.

Thirdly, to claim that the oppression women face is inevitably and inextricably linked to their biology enforces that very oppression by suggesting that there is something essential to womanhood that makes such oppression necessary or helpful to women somehow.

Fourthly, it's patently obvious to anyone who isn't engaging in bad faith that this is simply a reversal of terms to deligitmize "gender identity" language, without considering the context that makes that language meaningful:

She's right that she should get to present herself as she pleases, that she should get to think of herself as she pleases, and that others should respect that.

She's wrong that this somehow gives her the right to ratify or reject these same privileges in others.

The biological determinism she is advocating is what makes her experience as a quote unquote biological woman so difficult. She is actively fighting to maintain her own oppression rather than extend sympathy to others who suffer in other ways.
It is crab in a bucket bullshit.

She's unable to imagine the trauma of how it would feel to be forced to act like, and identify as anything other than the gender she performs because she feels that her biology has made those cultural factors irrelevant to her.

But here's the thing, if you can imagine a complex world of wizards and all that garbo, you can imagine yourself in the shoes of another person being forced to be something they do not feel they should have to be.

Lastly,

If she were not a cis woman, but rather a trans woman who looked and sounded exactly as she does... How does she imagine her experience would change? Would people submit her to a DNA test before being sexist?

If not doesn't that imply there is more than biology at play?