I'm a .375 fan, so if you can shoot the .378 well that's what I would choose. My bear rifle is a .short barrelled 375 Ultra which gives me 2600 fps with a 300 gr bullet. I think you will be happy with the Barnes TSX - I've found them to be much more accurate than the old X bullets.

On sheep, you might find that the X's won't open up as well as you would like. I played around with 260 gr Nosler Ballistic Tips and Accu-Bonds a bit, and the Accu-Bonds are the bullets I would choose for dall sheep. I found that I had the best accuracy when I loaded them to 2600 - full speed was 2900 - sub MOA at 400 yards. The lower velocity I think will give you better terminal performance, and less blood shot meat, yet these bullets are heavy enough to diminish the effects of the tricky winds you might encounter in the mountains.