carnist.cc includes typically a "which is it?" section, which can be argued against as a means of discrediting the claim.
You can just say that you don't believe the contradictory attacks on veganism. In fact, it might be absurd in some cases. For instance, "Animals will overrun the earth if we don't eat them" and "Farmed animals will go extinct if we don't eat them". It would be absurd to fully accept both of these claims.
But it doesn't really represent a cogent refutation of either of these points that you don't accept the other, and therefore this somehow bolsters your argument for the one you do believe in. Furthermore, you can always start qualifying and bargaining your claims. "Chickens will overrun the earth; cows will go extinct." Again, this isn't really worth a bullet point in discussions against the refutations to these claims.
None of this discredits that it's useful to get some perspective that these claims at the very least seem to have some compatibility issues. This is especially true for things like "Animals have no rights" versus "I'm against factory farming" where it seems that there might be some issue with simultaneously arguing that animals both do and don't have any moral value. This is something that most carnists genuinely have not considered.