She's not a conservative, she's a liberal (in the political science sense of the word, not the USian synonym for leftist).
It's not 100% clear where Rowling's transphobia comes from. She certainly fits into the group of transphobic cis women who have been abused by cis men and concluded that all men are evil, including the ones that want to be women.
But there's also a dynamic which I think you can see with Graham Linehan and Dave Chappelle as well. Born into comfortable middle-class families, well-educated, never really thought about their bog-standard liberalism. Became extremely successful, became accustomed to near universal adoration, made a thoughtless transphobic comment/skit, received criticism and reacted with absolute fury at the idea they could possibly be prejudiced about anything. Because they're liberals, you see.
All three just keep digging that hole deeper rather than face up to the idea that maybe they got something wrong. Linehan's career is over (as is his marriage), Dave Chappelle is hanging on by a thread and flirting with the right, and Rowling doesn't give a shit because she's a billionaire and does not have to give a shit about anything at all.