Since this is for Brown bear and you hand load I would pick the 9.3X62.

I would rather have the larger bullet diameter of .366 inches vs the smaller bullet diameter of .338 inches when going after brown bears.

Being that you hand load you can get some good velocity with heavier bullets. I think a 250 grain Swift A-frame or Barnes TSX at about 2500 - 2600 fps would make one heck of a bear stomper.