A few years back I was in the same situation. I talked to some guys about pistol shooting and started with the 223 Super 14" Contender with a 2x8 scope. I have taken PDogs out to 300 with it. The 222 Contender is a good round also.
I tried the 204 in the pistol length barrel and wasn't impressed with it. It didn't seem to buck the wind as good as my 223. I also got a 17HMR and here in Oklahoma the wind recked havock on the little bullets and sold it the day after the hunt. If you like to reload there are many out there.
I picked up a 222 Rem Mag in a XP100 and it has a little more gas at longer distaces than my 223. The 222 Mag is about 1/8 or so longer than the .223.
I also shoot the .243 which in the Encore will reach out past what you asked about with 80-87 grn bullets and is a thumper on coyotes. I like my 6TCU also.
I just got a 6.5JDJ but haven't had time to push it past 200yds yet. It would be a little over kill on PDogs but who cares.
I'm sure there are many out there as others on this site use but these are just the ones I use and like.
Heck, I took my 309JDJ one day just to fire form some loads. If I had some dogs that would just stick their heads out of the holes, I would just shoot at the mounds just below the top and blow the PDog out of the hole with it.
Go for it and have a good time. I also use my 22LR Contender when the dogs are brave and pop up close.
Mike