This is my first time publishing a strategy on Xu-Fu's guides. Please let me know if I made any terrible formatting mistakes.
There is a little wiggle-room in the first pet selection. I actually used Leap instead of Primal Cry and wasted a Black Claw on the round before I swapped in my Feline Familiar. While fine-tuning this strategy, I realized if I used Primal Cry instead, it would allow me to land a few hits first and save some HP.
(Point is, you can use any Beast pet in Slot 1 so long as it has Hunting Party and Black Claw, and as long as you follow the rest of the steps, everything should work out fine.)
Note: Please keep in mind that to complete the Achievement, all 3 pets must be level 25. Leveling options are included for alternate completion methods, but will not grant credit towards the individual or meta Achievements.