For just strickly paper punching, the .223 would be the most affordable (as far as factory ammo). Plus the barrel life would be longer than a 22-250.

As far as brand, everyone has there favorites. Since you mentioned cost, Savage would a good choice. Though the triggers need work. Other than that, I've found accurate and not so accurate examples of about every factory heavy barrel out there. I've had good luck with several Model 70 heavy barrels and a couple Remingtons as well. I haven't been super impressed with the few Ruger heavy barrels I've owned. The Sako and Tikka medium heavy barrels are super. The Tikka would defiantely fall within your price range.

I'd just look them over and pick the model that appeals to your eye and pocketbook.