My knee-jerk answer was a 25-06, something you can reach out a LONG way with, but if 200 yards is a long shot, most of the other calibers mentioned would do just fine. I've always hankered after a 257 Roberts, as well as a .243, but were I to go that low in caliber I'd get a 6mm for the longer neck and slight velocity edge. If a short action rifle is more appealing, the 7-08 is a great choice, even a 260, but it you want something different, and short, a 250 Ackley Improved would be COOL. It's the best of the Improved rounds (Per P.O. himself) and in a good rifle, with enough barrel, encroaches on 25-06 territory.
Just more food for thought. Isn't this fun?
Papajohn