I see this Terrakion + Whimsicott combo where you use Beat Up with Whimsicott, Terrakion gets Justified, and Terrakion sweeps. Now if Whimsicott held a King's Rock, and then used Beat Up on Terrakion, could Terrakion possibly get flinched?
Each strike has an independent chance of being a critical hit, as well as activating a King's Rock.
Bulbapedia’s page for Beat Up
Yes, king's rock makes it possible for beat up to cause flinch. I don't see why you wouldn't be able to flinch your own Pokemon in a double battle with a move that has a chance of causing a flinch. I don't know how useful this is though, are you thinking of tricking a kings rock onto the Whimsicott or something?
