I assumed from your post that you want something for the dogs, but the recommendations seem to be for deer. The .44, .45LC, etc. are all good choices for deer, but these calibers are way over the top for dogs and similar size critters, unless you're shooting them at 75-100 yards. Also, ammo for these cannons is going to cost you a bunch.
I would recommend a .357. You can fire .38 special loads, both for practice and for carry (yes, I would use a .38 v. .357 for the dogs). If you want something that can be used for both dogs and deer, check your state regs.