I go to battle the new opponents from the current plot only to find, "Oh no! None of your pets are eligible for battle. You might want to make sure that they're well fed and happy!" even though my battle pet Dobu's stats state, "Hunger: Bloated, Happiness: Very Happy and Health: 70/70"

Am I missing something?

Did you already check if your Pet is still stuck in a former battle? You can do that on the Statistics page of the Battle Coliseum - Click!

AHA! Thank you! For some reason clicking the "you lose" after my pet died didn't finish it out :)