Any of those four should be good out to 300 yards. The .338s will kill bears and moose cleanly with a decent bullet and good hits. I've only shot a couple moose and bear with a .375 Weatherby (like a 375 H&H Improved). Not sure I could tell any difference between it and the 338 Win Mag or 338 Imperial or 338 Edge.
If you're sensitive to recoil or you might use the rifle at longer distances in the future, the 338 might be the better choice. The heavier the bullet, the greater the recoil. The 338 bullets will have a little better Ballistic Coefficient for equal weight bullets. I've lived and hunted Alaska for the past 33 years, and I've ended up using 338 caliber cartridges for many moose and several brown bear. So I guess I favor the 338 caliber cartridges. But I've been known to take some pretty long shots over the years.
Again, if your range is limited to 300 yards or less, the 375 and the 416 should work every bit as well as the 338s - provided you can shoot them equally well.