I've shot more moose with a .308 than anything else, and it worked just fine with 165-180 grain bullets. I have also shot a few moose with cartridges such as the .35 Whelen and .375 H&H.
I prefer relatively tough bullets for moose, bonded, partition, etc. Not because moose are so tough, but they are big, and shoulder bones (Humerus/ scapula joint) are hard and can deflect or break up fragile bullets. Penetration is very important on some shoulder shots, and all rear angling shots, but almost irrelevant on side-on rib shots. Accuracy is WAAY down on the list of important considerations in a moose bullet. Don't choose a bullet that shoots like crap, but remember that Moose are real big, and are generally not shot very far away.
If I was going moose hunting tomorrow with my 30-06 I'd pick a 200 grain Nosler partition or Lapua Mega. If they weren't available I'd choose a 180 TSX, Partition, Norma Oryx, or perhaps a Speer Grand Slam, or A frame.
I have no personal experience with a .338 but the same bullets in 225-250 grain would do the trick.
