Despite all the hoopala about hunting BEARS, they are no harder to kill than whitetails of similar weight. Let that be a guide to cartridge selection.
I have only shot three black bears, but would have to agree with this statement based on my limited experience.
I shot two with a .284 Winchester with 145 grain Speer Grand Slams and one with a .375 JDJ Contender with 270 grain Hornady's. The all died real easy.
They are put together a little different than ungulates and I'd recommend studying their anatomy a tad before you head out. (If you haven't already)
I'd say you can't go wrong with either rifle, but in your situation I'd grab the .338 just to get it baptized....
Those 225 grain Accubonds shoot very well in my .338 WM.