Most moose taken by Alaska bushkins and folks outside the gravitational pull of Los Anchorage are taken with .30-06s .308s and .270s. My nephew whacked one we were chasing with a .30-30 a couple years ago. On the rare occassion someone is toting a magnum, it's usually a 7mm.
I usually use Federal High Energy with 180 gr Nosler Partitions, but they aren't needed. You will have satisfactory results with regular off-the-shelf ammo. The only reason I use them is because I also have a grizzly tag with me in case a target of opportunity should appear...........or wander into camp.
I wouldn't have any reservations about hunting them with a .257 or similar as long as I had premium ammo and bullets.
We also don't waste meat by taking body shots either.