I am receiving this message: "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 pet is fully healed and happy (10/10)

Is your pet stuck in another battle? I believe you can go here and check: http://subeta.net/games/battle/stats.php

thank you! it appears my pet was stuck in another battle. (: all solved!