A good heavy bbl .22-250 would suit you well, you'd be capable of much more than the 200yrds you mentioned, but you wouldn't be overgunned either. Or you might opt for a .223, finding cheap surplus ammo good enough to hunt to 200yrds isn't hard, it's nasty as all get out, and I'd never shoot it in any of my guns (except a $50 SKS in 7.62x39mm), but lots of guys buy target rifles in .223 just so they can get the ammo cheap, go figure, like putting 87octane in a indy car.