I have shot a lot of critters with a .308. I would not use a 165 Nosler ballistic tip on a moose or elk on a bet. But it IS a wonderfully accurate, ballistically superior and quick killing deer bullet.
Based on experience, I would feel confident using the following .308 165 grain bullets on elk or moose: Nosler partition, TSX, Accubond, and Remington Core lokt. Based on research I would also use the Sierra HPBT ( NOT their match bullet!) .
I have seen a couple of standard 150 grain bullets stop in the shoulder socket joint of elk and moose, and fail to penetrate the vitals. Even though they are quick killers if slipped through or between the ribs I don't feel comfortable using any 150's for big stuff with the single exception of the mono metal hollow point or tipped bullets like the TTSX.
If you are buying off the shelf ammo for hunting elk and moose, it is always a safe bet to buy 180's for your .308 if bullet construction is unknown or the type you really want is unavailable. Most any 180 grain bullet will work OK, most 165's will too, but 150's that consistently penetrate adequately on elk and moose from any practical angle are not so common.
There! that's all you need to know! ;-)