Let's suppose that this usage is in fact more common than the two that I cited as "correct". It seems to be either false or meaningless. What is Bob saying here?

You said in the OP that the more common usage takes the phrase to refer to any exception. So from that, Bob probably means that the brown bear you saw is an exception.

How does Rationalist Taboo help us?

Seeing as how Bob probably means that the brown bear is an exception, his argument is poor. So I would then say something like, "since you agree that there is an exception, you should agree that not all bears are black or white". If he disagrees, then he isn't using the common meaning after all and I would ask him to taboo the phrase "exception that proves the rule" to find out what he does mean.