No shit, Sherlock. It's an insult. Insults are pejorative. That's literally the point. It doesn't make it a slur, though. Slurs are about insulting someone for their genetic attributes, like their race or sex, or their sexuality, their nationality, or their religious or cultural identity, i.e. things that are inherent and largely unchosen about their identity. Slurs are not critiques for behaviors. "Asshole", "Fascist", "Bigot", and "Karen" are insults that are about behaviors, specifically about treating others without respect, equality, or basic human decency.

"Karen" is an insult for someone who acts entitled and who treats service workers poorly or sticks their nose into others' business and tries to police their behavior. The fact of the matter is that the majority of people that are that entitled and behave that way (in the US at least) are middle aged or elderly white women, which is where the name came from. But the term is not about insulting someone for being a middle aged or elderly white woman, is it? It's about their behavior. Older white women aren't the only ones that can be Karens, and most older white women do not behave like Karens.

If you cannot see the difference between insulting things you do vs things you are, you are probably very familiar with actual slurs.