To start with… I don’t like the term. It can’t be pleasant if you are actually called Karen. And I don’t believe I’ve ever used it….
But I don’t see why it is misogynistic…. Unless someone says that all women are Karens. Some women are just insufferable, the same way some men are. Admitting this doesn’t make you misogynistic! And more than calling someone a Dick makes that person a misandrist.
Bandying the word “misogyny” around whenever anyone says anything that’s not completely complimentary about a woman just devalues the word, and makes it more difficult to call out real misogyny.