Either that or instead have it increment a separate "N people have hit the Bozo Button for this user" counter.

Or, if you want it to also (or just) include it having a karma hit... make it actually carry a karma price for the user hitting the button too? ie, "I'm so sure that this user's a bozo that I am willing to sacrifice karma to say so, confident that I can earn it back far more easily than they can" or something like that? Since IF we wanted a "bozo button", we'd probably want it to count as something more extreme than simply hitting the downvote arrow. So we might also want to have a price attached to using it. At least, that's my initial thought here.