For most pets (except those two with healing) I use my "kamikaze" team
It takes 8 rounds for non-beasts, 6 rounds for beasts.
Tranquil Mechanical Yeti: Supercharge, Call Lightning and swapping him.
Darkmoon Zeppelin: Flyby, Explode
Pet Bombling: Zap, Zap, Explode
Zaps are not needed for beasts... they will die even with yeti's two attacks, flyby and explodes.
Dos-Ryga: Gilnean Raven, Crow, any fast or powerful moth (with Moth Dust).
Lucky Yi: Curious Wolvar Pup (with Snap Trap, Bite and Maul), Kun-Lai Runt and Stunted yeti (both with Rampage).