I have a 338 that pushes 225 grains at 2950 in a 26" Factory stainless spout on a Winchester Classic.... it is a pretty potent round for Elk, Bear , and Moose... recoil has never been an issue with the McMillan McSwirly stock. Great long range set up IMO...
Of the cartridges mentioned a big 7 pushing heavies would be a great choice also. In my view though, it wouldn't be a 7 RUM. I've researched the RUM rounds and find them to be too inefficient for the boat loads of powder you have to use to get them just a bit over velocities produced by more standard cases. And I know at least 3 guys personally who had 300 RUMs and got rid of them- too much recoil, extremely expensive ammo when you can find it, and even reloading for it was too expensive for the return on investment.