Too Dangerous for a Girl (a pretty great comics reviews blog, check it out), and it's taken me two whole weeks to come up with an answer. I mean, Super-Pets are AWESOME! Why would I want any of them dead even if I weren't an animal lover! If he's asking me to put down a member of the Legion of Super-Pets specifically (which is how the question could be interpreted), I can't see any of them deserving it. If I cast my net across all my comics lore, it doesn't get any easier.
There's the meta-continuity answer, I suppose. I don't care for the current incarnation of Krypto as a Game of Thrones dire wolf, but Krypto as a pure concept is usually wonderful, even if I've been very public about being a cat person. I don't have an affinity for dogs in real life, but they make great fictional characters.
So let's just take all the cats and dogs off the table. Even less pettable animals, like Bat-Cow, Throg or Topo, are clearly too cool to take to the vet for that final "exam". Heck, I even like Bill & Don the lobsters from the Fallen Angels mini-series! I actually think there should be MORE Super-Pet action in today's comics, not less!
Do anthropomorphic animals count? No, forget it, there isn't a member of the Zoo Crew, an Amazing Friend of Spider-Ham's, or a denizen of Aw Yeah Comics! I would want dead. So even if they counted...
So do I go with "annoying" and it's "goodbye Gleek"?
DECISION TIME! Ok, so I decided to go with a Super-Pet I didn't care about personally, that had not pettable characteristics to make it endearing, that was a heinous idea from its first appearance in 1975, and was further ruined in the 90s. Who am I talking about? Why, Green Lantern's pet, Itty, of course!
THERE! Does that satisfy you blood lust, Master Gray?!