My battle pet has 10/10 health and happiness, he's not in training and has weapons equipped but I get notified that none of my pets are available for battle and that I should check their health/happiness.

I'm having the same issue with my pets as well. It seems as though the battle coliseum has been extra problematic lately. :(

[edit]

Turns out my pet was already in a battle. Check to see if perhaps that's the case with yours as well?

here is the link to check if your pet is already in a battle btw :)

are your pets healed? check here

did this clear up?

Yes it cleared up--thank you all!