I think it's deeper than that. Klyden exists to represent Moclan society as a whole. He is the stand-in for their traditions, world, history, and culture.
We, the audience, are presented at the onset with a society that is male-only. The ship's crew, along with us, are sort of hand-waved away when asking questions about how things work in the bedroom, but on the whole, no one seems to have a problem with their culture. In fact, we even see this male-only species reproduce successfully before we learn that there are the potential for female infants.
In Moclan society, being born female is an aberration. It's not a biological necessity, and, for whatever reason, the Moclan culture views "being female" as a birth defect, one that can be easily corrected. It's sort of how, today, we view children born with a clef palette. There's no good reason to keep it around, and lots of reasons to repair it as soon as possible. Klyden represents this mindset and viewpoint perfectly.
Imagine someone fighting tooth-and-nail to not repair a cleft palette, or some other easily-fixable birth defect. Imagine them standing up in court and declaring that this obvious flaw is something that no one has the right to fix. Klyden is, from his own experience, outraged, and furious. Put yourself in his shoes, and his actions have nothing to do with bigotry, or hate. He's not angry at his child for being female, or at his husband for supporting her decision to become female. He's mad at the world because his entire world-view is challenged by his family.
In fact, he sees his culture, history, society, and even legal system saying that he is right, that the child should be male, and then he sees his husband and child, serving on a Union starship, talking nonsense about a "choice." That line where he says he wishes she'd never been born wasn't anger at her. It was anger that he is being forced to choose, and no matter which thing he chooses, he will loose a huge part of himself -- either his family, or his history.
And if he chooses his family, he has to confront the fact that what was done to him was just as wrong as what he did to his daughter.
Few people, even space aliens, have the emotional maturity to handle that kind of revelation in the moment without doing something regrettable.
But fuck, this kind of novel is why I love this show so much. When was the last time you had a long talk about that time Riker killed all his clones?