Ammo cost is meaningless when talking 100-300yd accuracy. You are loading your own, right? If not, 308.
If you are, the key is bullet and barrel. 22 has real br quality flat base match bullets available. Get those. Find a rifle with a heavy barrel and a slow twist like 1:10 to 1:12.
Get a Weaver T24 or T36. 24 would be easier to shoot. You might even look at a Super Sniper 16x or 10x. 10x is a tad weak for best accuracy, but usable.
Get the high end Redding die set with micrometer bushing neck sizing, body die, micrometer seating die.
Anyways....yes, 223 bolt rifle like a Savage 12 BVSS or 12 LRPV. 1:9 twist