Man - good luck to you. Wish I could afford a brown bear hunt. I have a friend that took a nice one with one shot from his 300WSM. And I watch a video of my cousin shooting one 4 times (2 in each shoulder) at 75 yds with a 375 H&H. I guess it just depends on the bear. But IMO - the bigger the better and your choice of a 375 Ruger should do the trick.
If I ever do go I would probably take my 325WSM.