DOGGONE IT MT, ya beat me to it.
I was going to suggest going to the .338 Win. Mag. myself.
However, since the question is about .300 mags., I go along with the suggestion of the .300 Win. Mag. I have shot the WSM in two Winchester M70 rifles, the one with the Featherweight stock and the one with the more conventional stock. I see no real difference in recoil from the Featherweight stocked version and a normal .300 Win. Mag. The one with the conventional stock kicked about like a lightweight 7MM Mag. to me, but I felt it was a bit too heavy to be lugging up an 9,000 ft. MSL on an elk hunt.
Just as a matter of comparison, I have two Winchester M70 rifles; one a .300 Win. mag. and the other a .338 Win. mag. Weight of the two is almost the same with the .300 being a few ounces heavier. Recoil between the two rifles will surprise you as when both rifles are shooting 200 gr. bullets, the .338 kicks less than the .300.
Of course, as bullet weight goes up, so does recoil in the .338. You just can't get away from the laws of physics. Still, if I were to end up with the money for a good elk hunt up north or a moose hunt for that matter, I would be taking both my .338 Win. mag.s loaded with a good stiff handload and 250 gr. Nosler partitions and I wouldn't look back. In really good elk and moose country, there is always the possibility of an encounter with "Brer Griz" and I'd feel a lot better with the .338 Mag and the heavy bullets.
The advantages of either the .300 Win, mag. and .338. Win mag. are you're more likely to find a source of factory ammo at come country store should you lose your loads than if you went with the Weatherby or one of the Ultra Mag types.
That's my take on the matter.
Paul B.