People really ought to learn that pointing out a fallacy doesn't suddenly make it false. I mean I could point out that that is known as the fallacy fallacy, but then I'd be doing the same thing myself.

Really a fallacy represents a particular weak argument. The way to defeat them is not to name the correct fallacy but to point out the weakness.

If that doesn't work then you likely have the wrong fallacy, or the argument isn't fallacious to begin with.