(For the record, this is veering off the topic of what jokes are appropriate in PPC areas. I'm not intending this as a suggestion for Board/Discord policy, just trying to explain my perspective in general.)
So, yes: the behavior of Karens, not the race of Karens, is the problem.
However, I don't think you can divorce the matter of race from the behavior in this case. The fact of white supremacy in this country is what allows Karens to think they can get away with victimizing Black people. If there are non-white Karens and they're doing the same thing, they're benefiting from the same systems of oppression, so I can't say that I mind lumping them in.
Being white doesn't make you a bad person, but taking advantage of the privilege afforded to you by whiteness in order to harm others does. As the behavior of Karens is symptomatic of the larger problem of white supremacy, so I believe the race of Karens is absolutely relevant.
Ultimately, I feel the distinction between making fun of white people and making fun of others lies in the fact that whiteness in this country is used as a means of determining who holds power and authority. I believe those who hold power must be subject to criticism, including ridicule, to prevent them from wielding their power unchecked. This is why freedom of speech and freedom of the press are so important—they're supposed to help guard us from tyranny. As things stand now, whiteness is used as a determiner for who is allowed to hold power. As long as that's the case, and as long as white people are disproportionately advantaged while other people are disproportionately disadvantaged, then I believe white people can suck it up and take their lumps when jokes are made about whites.
^ See? Nuance. Nuance means there may be exceptions to the general rule depending upon circumstances.
Incidentally, if anyone is inclined to cry "not all white people!" here... Everybody knows it's not all white people. But it is most of us, if only through complacency and silence. I include myself in this. I am often complacent in my privilege, and I could do more to stand up for others. When I do nothing, I am part of the problem. But, recognizing the ugly truth behind the jokes is one small way I'm trying to be better, and sharing these thoughts where they may have some impact is another.