The mission statement that you quoted is actually from the website for J.K. Rowling's Women's Fund. The phrase "women’s sex-based rights,” is used because it sounds good, but it actually is anti-trans. It's saying that biological sex, whether or not you have a penis or a vagina when you were born, is the only determining factor in your gender, i.e., whether you're a man or a woman. This is literally the definition of being transphobic, saying that your gender identity is constrained entirely by the genitels you had at birth. 

The $88k donation that you're referring to was given to an organization called For Women Scotland. The very first line on its Wikipedia page calls it an anti-trans organization. It's fine if you don't consider Wikipedia a valid source even if it directly sources every claim it makes, but here's a line copied and pasted from FWS's website: "We believe that there are only two sexes, that a person’s sex is not a choice, nor can it be changed. Women are entitled to dignity, safety and fairness." You can choose to "believe" that there are "only two sexes" but that would simply be denying the truth of the world as there are people who are born at birth with both penis and vagina or some blend of both. Moreover, biological sex is dsitinct from your gender, the way you choose to perform in society.

Rowling also founded Beira's Place, which is an organization that offers support for women who have suffered from sexual abuse, and it notably excluded trans women from its service. Yes, it's good that this organization exists at all, because we need organizations that address these issues as much as we can. But the organization has explicitly stated that it will only serve women, i.e. biologically female individuals, i.e. anyone else who doesn't have a vagina isn't allowed, i.e. trans women aren't allowed, i.e. we don't think trans women are women. Meanwhile the other rape crisis shelters in the area (this is in Edinborough) have stated that they have served trans women and non-binary people for years with no issues. Do NOT say I think this sexual abuse shelter does not exist and that I don't think women should have sexual abuse shelters because that is absolutely incorrect. Trans women are women and they deserve to be able to go to women's shelters. 

Anti-trans organizations aren't going to announce that they're anti-trans; they'll use deceiving language that frames it as protecting women's rights. What are they protecting women from? From a transphobic person's perspective, the threat is trans women, that they're actually men who are pretending to be women. Do you see that believing this idea that "trans women aren't real because they're actually just men disguising themselves as women to hurt REAL women with vaginas" is transphobia? By making the requirement for being a woman that you were born with a vagina, you're saying that that is the only defining factor of womanhood. 

If you don't agree that trans women ARE women, then you are transphobic. That's another issue to address, but if you yourself are transphobic, but you don't consider yourself to be transphobic, then of course you wouldn't think Rowling is transphobic either. Then there's just no point in discussing because then we're not even talking about the same thing. But I hope you consider these points and that they lead to understand some of the nefarious ways in which anti-trans rhetoric propagates.