With solid copper bullets you normally drop down in weight. Normally I'd say 130's are about right for a 308, 150's for a 30-06 and 165's should normally be used in magnums only.
Your situation is a bit different. Any of the weights should shoot fine, for paper punching. If that is your goal, then shoot the one that proves more accurate. Just a guess but 130's or 150's would probably be best. Twist is related to bullet length, not weight. The copper bullets are longer than traditional bullets of the same weight.
If you want to hunt with it then 130's or even 110's. Barnes bullets need a lot of speed or they won't expand. You cannot get a 165 gr bullet fast enough from a barrel that short to get reliable expansion at 300 yards. Even in a longer barrel only at close range. Even with bullets as light as 130 you will get better penetration than from a traditional 165 gr bullet. On par with a traditional 180.
