I randomly picked an SS Grove Viper as my first pet, used Hiss on Bucky on a whim as it was faster than Bucky, then proceeded with the strategy as normal -- but Netherpup was able to finish Bucky off on its own. Awesome strat! :)
Son of Skum can almost solo the fight. Make sure the other two pets are set up so that Skum is the pet switched in on the first turn, and then use Lightning Shield and Stampede. The damage from Bucky's stampede will nearly kill him by the time he's taken Skum out.
I would advise taking the Diemetrodon Hatchling (or any other Beast with Quick Attack) as the first pet in this strat. First, you get to do something more constructive with your first move than pass, and finally at the end of the battle having that Quick Attack on the back of Netherpup's Howl buff ensures you have less to worry about with RNG.
Kid, Anklerender, flat tooth calf (or anything with a big 1 turn attack), stampedes + hunting party for the win (use primal cry instead of black claw, give you two attacks from hunting party and double dmg that you won't otherwise get for the headbutt)
my apologies for above post i was TRYING to use runt but it wasn't working.
Simple 4-5 turn rotation.
Spot 1: S/S Summit Kid XXX as it will get swapped
Spot 2: S/S Zandalari Anklerender 2-2-2
Spot 3: S/S Zandalari Kneebiter 2-2-1
Round 1: Summit Kid - Pass, will get swapped out
Round 2: Anklerender - Black Claw
Round 3 -4: Anklerender - Hunting Party - Anklerender dies
Round 5: Kneebiter - still 1 round of Black Claw remaining, 1 round of Hunting Party kills Bucky FTW.