I would choose the 338 Win Mag. . The 300 H&H is equal to the 300 WM and here's how it compared with my 338 WM.
I can't imagine being limited to only two rifles, but if a hunter wished to limit themselves to only two rifles, for hunting in North America, I believe the 308 Winchester and the 338 Winchester Magnum would be the two to choose.
My situation for an early fall elk is different. . Can't take'm all. . At first I was going use my 300 H&H and loaded up a few boxes of ammo to sight in a new scope and get all practiced up . . I took two rifles out to the range, my 300 H&H and my 338 Win Mag with the idea I would shoot the 338 while I waited for the barrel to cool on the 300 H&H. . The 338, with a 200 gr. bullet with 74 gr. of IMR 4831, was equal to shooting the 300 H&H with 180 gr and 68 gr. of IMR 4831. . The recoil on both rifles was pretty much equal, but I realized for an Elk hunt the 338 WM is the better choice. . Both rifles grouped nice at both 100 yd and 200 yd,and the velocities are not much different between the two, but the 338 WM\ I believe is the better choice .
I went again and increased the 300 H&H to 69 gr of IMR 4831, with 180 gr bullet, and I noticed the pressure was up as the bolt was just a tad hard to lift. . The 338 WM I increased to 75 gr. of IMR 4831 with a 200 gr bullet, and the bolt opened easy and the grouping remained tight. . The 300 H&H, at 69 gr, showed a variance in grouping. At 68 gr it held tight.