I just beat Carrot. It wasn't all that bad until it's health got low, and it wouldn't lay off the Regen and Protect. I just kept Balthier at a distance to keep him from getting hit by Bad Breath, and had him use Remedy whenever someone got hit, since his Remedy removes all of the status effects of Bad Breath. Once it got too low to bash down, I just hit it with a Concurrence, and no more Carrot.
Edit: I tried setting up a Dispel Gambit on Panelo (White Mage/Black Mage), but it couldn't keep up. Probably would have worked, eventually, but I chose the "You die now!" button.