Re: Which of these bullets would you use in a 338RUM?
For long range shots select a bullet with a BC on the high end that shoots very accurately out of your rifle.
For bear protection, a camp load, or close range shots (whatever you decide is 'close range') use a 225 grain Trophy Bonded Bear Claw. If they still make them like the ones I have used on two different mature boar brown bear on Kodiak Island, they are deadly.
One +1000lb boar simply broadside chest shot was instantly down and dead from 130 yds. This bear was unaware of my presence, so the adrenaline wasn't pumping.
The other boar had already been hit and was making a get-away - fully adrenaline charged. Same shot from about the same distance. The bear was rolled off his feet and never regained all fours - simply center of the ribs. This tells me this bullet was creating a tremendous physiological shock to the animal, sufficient to outright lights-out kill, or disable these bears.
Trying to find a bullet that excels for both close and long range use generally will result in some compromise from the ideal bullet for both extremely different purposes. The long range bullet selection is by far the more critical bullet selection. Just about any premium controlled expansion bullet will be accurate enough for the closer range work, and these bullets are constructed to withstand high velocity impacts associated with these close shots. I simply mention the Trophy Bonded Bear Claw as it proved deadly with both brown bears I hit out of my 338 Imperial Magnum, which is the same as your 338 RUM but with the shoulder shoved forward about 0.050" farther for a little bit more case capacity.