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Keep in mind, the single-pet fights, (Unbreakable, This Little Piggy, Mind Controlling Plants, Crawg in the Bog, and Jammer), DO NOT count for the Family Battler Achievement. This strat was made before I knew that.
I don't know if any of the other breeds work for this, hence the breed restriction.
I think we've established that "Ca-Caw Ca-Caw" and "Tookie Tookie" don't work.