I don't think the dead elk or moose will complain either way.

Personally I would choose a 180 grain bullet for the 300. A 180 grain premium bullet out of a 300 will cleanly take out any elk or moose with no problem what so ever.

If I were to go with the .338 I would then choose a 225 grain premium bullet. Again, this would be able to cleanly take out any elk or moose you will ever encounter.

Both of these choices are heavier than average bullets, but not the heaviest, to give you and plenty enough "power" to break big heavy bones good penetration after those bones have been broken. IMO they will provide you the best of power and flat shooting ability.