"Stopping power", if it does exist, is affected by caliber. Taylor's KO charts, for example, gave better "stopping power" (KO values) to larger diameter bullets. Makes sense, if you think about it: with a given amount of energy, the larger frontal area of the larger caliber projectile will deliver more energy and penetrate less as it strikes a soft target, the smaller diameter projectile will deliver less energy and penetrate deeper. Kind of the "5.56 vs 7.62" internet argument you see a lot of.
"Hydrostatic shock" is mainly a function of velocity, so perhaps a smaller diameter projectile would have an advantage
since it does not slow as quickly. I don't know about that one, I would have to think about it for a while.