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Any 165 grain hunting bullet will kill any game you care to hunt. Since you're not reloading, you'll have to try a box of as many brands as you can to find the ammo your rifle shoots best. Sight in, then practice, off hand, at 100 yards, on a 9" pie plate until you can hit it every time.
Gosh premium ammo is expensive compared to corelokts
I recommend 180 grain soft points such as Corelokts or the Winchester/Federal versions.
If you feel you can afford premium ammo I would recommend a load be it Winchester or Federal shooting the barnes TSX bullet or the Nosler Accubond or the Nosler Partition or any other bullet that is bonded or has some design to hold it together.
Moose and deer are not particularly hard to kill and 150 grain bullets will certainly do the job with less recoil. It really isn't going to matter what you use.
165gr Federal Fusions. Best factory ammo for the buck I have seen yet. Accurate, high weight retention with excellent expansion don't expect super penetration like a Barnes or Partition, but it'll do.
I have shot many deer and a handful of moose with my 308 150gr. handloads I haven't fire a factory round a game in 35+yrs but if i had to i would use a premium 165gr in a 308 win.
The .308 is not at all "hard" on bullets, in terms of demanding premium projectiles - at least relatively in terms of comparison with the .300 Magnum and the like.
150-180 grain factory bullets should do fine for the hunting you describe.