First, once you get your taste of long range P-dog splattering, you won't limit yourself to 200 yards.
I have shot P-dogs with 6mm WOA, 22-250, 223, and 204 Ruger. All four work well, but the 6mm WOA really shines in the wind, at longer ranges. In my shooting, the 22-250 and the 204 Ruger are very close to ballistic twins. I use 50 gr. VMax bullets in the 22-250, and 39 gr. BKs in the 204 Ruger. I might have to give the slight nod to the 204, in the wind, and less recoil helps to stay on target to see the hits...Or misses. I also shoot the 50 gr. VMax bullet in my 223 Rem. P-Dog loads, which has some pretty spectacular hits at ranges I shoot the 223 Rem.
So here is my current P-dog rifle battery: 6mm WOA(AR-15), 22-250(Howa Heavy Barrel Bolt), 223(AR-15 Wylde, Tikka T3(1:12 twist)). My 204 Ruger was a Savage 16, but has been converted to a 260 Rem, and I am in between 204 Rugers at the moment. I am waiting for my rifle funds to replenish, and I will be assembling a new 204 Ruger upper, or a complete rifle, depending on the rifle fund.
My recommendation, is to start out with a HBAR profile AR-15, in either 223 Rem.(cheap available ammo), or a 204 Ruger, if you want to grow into longer range shooting. The 223 Rem. option is the cheaper way to go, both in firearms cost, and ammo cost, but if you see yourself trying to flip P-dogs, at 500 yards, in the future, go with the 204 Ruger.
Squeeze