as of now, no. if and when (and that's probably a big if haha) there's ever another clearing of battle logs, which would mean we could actually optimize the battle logs table with good indexing and such, it would be a possibility.

if you're in already a battle there won't be a dropdown at all, but a link to the battle you're already in (since you can't be in two battles at the same time with different pets, either).