Use heavy lead!
I wish I had the opportunity to hunt a large brown bear! I have been on a few black bear hunts in WV, but that's a different ball game. (we use 30-30s and 357 to 44 magnums) If I was hunting brown bear, I would use a 405gr flat tipped lead 45-70 round. That cartridge was able to take out buffalo in the 19th century at several hundred yards, so I imagine it would work great on a brown bear.
I base this off of the heavier bullets' kinetic energy, and its ability to carry momentum through the thick flesh and bone of such a creature. Also the heavy lead doesn't break apart like hollow points. You're wanting deep penetration/slow expansion. I would also refer to the style of bullets used in Africa in the 19th century as well. What works, works. History proves that, so why reinvent the wheel? My 2 cents anyway...