I'm sure that a 6.5X.300 Wby with the proper bullet would kill a brown bear, the question in my mind is, How long would it take. Those bears can cover a lot of ground in a very short period of time. Looking at your battery I see many more suitable arms for this task. A .300 with premium heavy bullets and a .340 Wby would do. To me picking out a rifle I would choose a .375 Weatherby, probably for nostalgia sake.
If you're looking to be the first to bag a brownie with a 6.5X300, go for it but make sure you have a reliable guide or back up with a boomer. You don't want to end up being a tasty snack, do you?