I've been party to killing 5 browns and 1 grizz. I used 225 grain Barnes Xbullets for four of em and the 225 grain Barnes TTSXs for the last two all in 338 Win Mag.
It's a tradeoff and you'll be fine with any of em. The monolithics like Barnes will generally penetrate deeper than the controlled expansion type bullets of similar weight like the Swifts and Woodleighs. This is because the monolithics make a smaller wound channel than the others. The frontal expansion of the Swifts etc. will make a wider wound channel but will penetrate a bit less. I doubt either design will fail you.
I personally have no need for bullets larger than 225 grains in the 338 Win Mag and have hunted with this combination exclusively since 1993. Almost all my experience is here in Alaska as I have lived here since 1985. Uncle Sam did move me out of state and back twice during my military career but the 338 Win Mag and 225 grain Barnes bullets have performed well for me on blacktails, caribou, moose, goat, black bear, browns and grizz. You'll have no problem with the 340 and any of the bullets you have chosen.