The 17 caliber bullets are affected greatly by cross winds. Maybe a 22 cal cartridge? You don't need the 22-250. I love the 223 Rem., (you should be very aware of this cartridge as you are military. I use a Bushmaster Varmint rifle in 223. Very accurate)
The .308 is a great cartridge. By buddy just shot a big cow moose (650 pounds, field-dressed) here in Maine with a .308. He used 150 grain bullets. He shot 3 times as she ran across a potato field. All three shots were at 150 yards and complete passthroughs. (He shot 3 times because she was running straight for the game preserve border)
Kev