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given your libertarian bent, I worry there’s some overlap

I doubt it. I don't know much about JK Rowling's politics other than a few statements about trans women. But here are my takes on those things you mentioned:

• gendered sports - government shouldn't be involved, other than a blanked anti-discrimination law; if a school or sports league wants trans women to play on the women's team (or trans men on the men's team, but that's rarely an issue), that's totally fine; I think the only reason a sporting organization should ban people from participating is if there's an unfair advantage, and I haven't seen any evidence of that for trans women since it seems hormone therapy kills most of the physical advantage (other than height)
• bathrooms - I really don't understand people's obsessions w/ bathrooms; I wouldn't mind the opposite sex using my sex's bathroom, provided they're not creepy about it; I'm a parent with kids of different sexes, and it's never been a problem bringing them w/ me or going w/ my SO into the different bathrooms; the only law here would be not accommodating people who need to use the restroom (i.e. if one is full, people should be able to use the other), but private businesses should be able to choose their own policies
• sexual abuse shelters - I don't know much about this, but I'd absolutely be happy to donate to any shelters that need funding; they tend to not advertise where they're at, though, for obvious reasons, but I'd have no qualms supporting a shelter whether they support trans people or not, provided they're doing a great service for those who need them

I really don't care if someone chooses to be treated as the opposite sex, and I'll call them by whatever pronouns they prefer, because I'm not a jerk. We have a trans woman at our local library, and the only reason I noticed is because they transitioned while we were routinely going there, and I made the connection that the man who used to work there is probably the woman who now works there. I'm totally happy using whatever terminology trans people want, I just disagree that there's some innate concept of gender, but that sex is the only thing that exists in reality (chromosomes and anatomy don't change, even after surgery), and that gender roles are largely invented (though inherited from physical characteristics; e.g. women are child nurturers because they were the ones who produced the milk for thousands of years).

I think it's largely a distinction without a difference. I probably treat trans people similarly to how you do, I just disagree about what "gender" is. I think it's an idea we've largely invented to describe social roles, and in our modern age, those roles are a bit more fluid. Some women are the sole breadwinners of their family, and some men are the child rearers. Some women are absolutely shredded (like the woman in the picture above), and some men are very effeminate. I think the real problem here is that the terms "man" and "woman" have become so loaded, so much so that people don't feel comfortable identifying as the label that matches their anatomy.

Honestly, I think it's completely reasonable that people would prefer to switch genders rather than fight to fix the assumptions and whatnot that we have associated with gender. So I absolutely support trans people because sometimes the partial fix is more than sufficient, and I support any policies that give people the choice to identify however they choose, provided that doesn't provide them with an unfair advantage in some way.

So yeah, I don't think the reason matters as much as you say. It's completely reasonable to believe something to be true while supporting something that seemingly goes against it for other reasons. I believe in maximizing individual liberty as much as possible w/o infringing on others' rights. I think everyone should be able to live the way they choose, provided they don't interfere w/ others doing the exact same thing.

Respect each other, and fight for them to be able to live how they choose. That's more-or-less my political philosophy.