Teatime's first appearance hints that he's the perfect person to take out Not-Santa. The very first thing he does on the page? Sneak into someplace undetected through the chimney.
When Violet tells Bilious she's a member of Offler's League of Temperence, Susan reasons that it makes sense that Offler's followers would forego alcohol, as crocodiles don't go into bars often. There is a fairly well-known joke about a man walking into a bar with a crocodile under one arm, but considering how it's not a drink that's put in the reptile's mouth, it makes sense that it wouldn't want to go there voluntarily...
The not-so-Good King Wencelas whom Death-as-Hogfather confronts is seen forcing a profoundly embarrassed and ashamed woodcutter to accept some Hogswatch charity in the form of the leftovers from his feast. In other words, by "giving" something in a way that causes such distress in the recepient, the king is committing an anti-crime.
The Analogy Backfire when Teatime declares himself to be a guard's worst nightmare is framed as a joke, but in hindsight it makes some sense for him to take Teatime literally: that's actually what the tower's defenses are.