>> The impact of the bullet is exactly the same as the recoil of the gun; if you can take it, the dog also can.
LOL, that’s a ridiculous description. You must account for distribution and inertia.
Both objects (you and the target) are at rest, for all practical purposes. You only need to account for distribution. Since the bullet is smaller, it creates higher pressure and can penetrate.
In other words, imagine that two people are in a duel. Person A shoots and hits the barrel of the B's gun. The bullet enters the barrel and stops. Both persons feel exactly the same impact. You can generate additional energy only if you fire rockets.