I hunted Brown Bear in 1996, I used a 375 H&H with a 300 gr bullet 4 shots at about 125 yrds and Backed up by a 338 at 4 shots . Unless your are very lucky to put the bear on the ground with one shot Bigger is Better ,but the bullet must be placed where is is supposed to go in the vitals.
My buddy shot his at 110 paces 9' 3" 4 rounds from an 06 backed up buy 4 Lightfield slugs out of a TarHunt slug Gun. The bear is pictured on the Web site for Tarhunt.
I will do it again but with a 416 Rem mag with a 400 gr bullet.