Being a Karen has nothing to do with your actual first name.
What you're really doing there is showing why it isn't sexist even though it's implicitly female. The detail is only there for color, not as a core component of the archetype.
"Karen" is rooted in the stereotype of an over-bearing, entitled person in a customer service setting. Our prejudices tell us that person is probably a white soccer mom. So she gets a name we associate with the white soccer mom demographic.
You argue that "Karen" doesn't refer to whiteness, the middle-classes or the female gender - it just refers directly to the stereotype and therefore it's OK.
I invite you follow the same logic here.
Consider the stereotype of a gun-toting street gang member. Our prejudices may tell us that person is probably a young black male that lives in a big city. So he gets a name we associate with the young black urban male like "Deshawn" .
The logic is the same, yes?
If you're OK with "there goes Karen calling the cops on the Walmart clerk again", you should be OK with "there goes Deshawn shooting the neighborhood up again".
It is insulting via the sexist, racist, and stereotype routes simply because you meant it to be insulting. You explicitly use the "image" of a white woman in a stereotypically unflattering situation as an insult. In this case it's your implication that [quoting you] "a specific type of asshole" must be associated with the image of a white woman as represented by the name "Karen".
Only a serious level of bigotry would make someone pretend it's not racist or sexist if the word can also mean something else, in this case a simple name. It's what you meant by it and what you associated that with that matters.
No my friend, it's also insulting via the "why you picked Karen (implicitly white woman) as a reference for an insult" route. You may use the insult on anyone else but the fact that you picked "white woman" as the reference means you believe that this is a defining enough trait to associate with them, you consider white women the embodiment of that negative trait and as representative of it.
It's the same reason calling someone a "fag" is meant as an insult for both that person (even if they are straight), and for homosexuals since you obviously consider this trait of theirs only suitable for an insult. It's an attack both on the people you address it to, and on the ones you based your insult on. Something an uneducated bigot would readily use but never realize why. They usually feel threatened by everything they don't understand or like (different gender, color, sexual or religious orientations) and "weaponize" that as a cover for their own ignorance. ;)
I would dispute this point. The term is, at least, derogatory to women who are named Karen -- whether or not they are "Karens".