Ugh. Okay, no I didn't figure it out. Why does my pet stop appearing in the Battle Coliseum's drop-down list after a while? I'm not stuck in any battles and his health is fine.

Is your pet full and happy, too? Check that first. If not, feed and play with them before trying to battle. If they are, try clicking the get unstuck from battle link anyways. If it still doesn't work, you can file a ticket :)

I'm experiencing the same thing. P: My pet is full and happy too.

Off to make a ticket.