The 338 used to have factory loaded 300 grain softpoints. I really could not tell the difference between getting thumped by my 338 and my buddies
375 H&H at that heavy a bullet. I am sure the 338 would do the job, but I am the type that considers fatter calibers for bigger critters. Thus I would go 375 and the ammo is not anywhere near the cost of 416 Rigby. (Heh, some folks also use the 7mm Rem Mag on bears.)