Isn't it when they're not fighting they start regening? I know when I used Stay then they just heal fast. But if they're just following you or using heel
I dunno about any of the new jugs or job traits or whatever, but I CAN say from personal experience that this guy is right^ All pets, even charmed ones, will regen when they're not fighting. I know this because when I was just starting out on BST, I didn't know Stay was used for anything other than running the hell away from the mob before it de-charmed (pre-leave, obviously), so would usually just stand around waiting for the darned thing to sloooowly get health back (and recharming every so often) before siccing it on a new mob.
Thank goodness I eventually found out what Stay was really used for -_-