Dakor,
I have only had 1 experience with alaskan brown bears, but it was up close and personal. My buddy shot a brown bear at 60 yards with a 30-06 shooting partitions. I knew he hit him, but it was darn hard to tell by how fast that bear took off. This was his first time to ever kill anything and he got a little rattled on the follow-up shot. Rather than let the bear reach the brush, I thumped him in the shoulder with a .375 H & H shooting 300 grain partitions. Looked like you hit him with a truck. Bear went down hard.....but, like Brent said earlier, be ready for that bear to keep going. Bear jumped back to his feet and took off again. We both shot him again and he stayed down. Autopsy showed 2 from the 30-06 right behind the shoulder and 2 from the .375 in the shoulder. NEVER underestimate these bears ability to take a hit and keep going. After the shooting was over, we both crept up to the bear, guns at the ready, and gave him the ol poke him in the eye to see if he was dead routine. He was. We put our guns down and my buddy straddled him on his back and tried to pick up his head. His grip slipped and he sat down on the bear to get a better grip. When he did, I guess the bear's lungs had filled with air because when he sat down the bear grunted. For a split second, my buddy had a look of pure terrior in his eyes as he thought he was sitting on a wounded, but live bear. I just about broke my ribs from laughing so hard at the look on his face.
In summary, get close. Don't take a 300 yard shot. No matter what you shoot him with, expect to shoot him again and be ready to do it. You didn't say what your brother will be carrying, but I wouldn't personally buy "just" a .338 for the hunt. Depending on the back-up gun, I would either stay the course with your 300 WM or go on up to a .375.
Have fun on your hunt!!