I've discussed polar bears with a few natives in Alaska and also in the NW Territories. One in the NWT guides for them and has supposedly been in on around 100 polar bear kills. They all seem to prefer mild mannered rounds of .257" or 6.5mm. The NWT guide recommends the .243 Win to his clients. They all say that .300 Magnums (I presume they're referring to the Winchester round) is too destructive. I am not sure if they'r referring to the hide, the meat or both.
I am guessing they shoot their bears bayed up by dogs or those chased down on snow machines, so close shots are the norm...but that is just a guess. Take that for what it is worth...
Money is a big factor. The last time I hunted with Inuits, a "summer" hunt for musk ox and caribour along the southern shore of the Arctic Ocean near the mouth of the Ellice River, my primary guide wouldn't even use such a powerful rifle as his .30-30 Model 94 Winchester if he could help it, because it made too big a hole in polar bear hides--which at the time were bringing up to $2000. This was around 20 years ago.
From our conversations (which were translated and added to by his nephew) most polar bears were either taken by using sled dogs to bay them, or by stalking amobearsng the pressure ridges of sea ice, which allowed the hunters to get well inside 100 yards.
