I can't seem to battle with my main pet, despite the fact that he is not in the training center, has full health and is not hungry. Is this a bug, or is there something I am missing?

Did you check he's not still in an unfinished battle?

Ah, that was exactly it. I hadn't thought about that. Thanks for the help!

No problem :) I had the same issue battling a while ago.

it doesn't matter if your pet is in training center or not. All of my pets are constantly in training. They can all fight.