I can't challenge any opponents due to the, "Oh no! None of your pets are eligible for battle. You might want to make sure that they're well fed and happy!" message. Yet I know at least one of my pets has full health and happiness, so what can I do?

Did you check the statistics page in the battle coliseum? You might be stuck in a battle.

...Turns out I was. Oops. Thanks! :P

Locking this thread since it was resolved. :)