First, if you do a search, there are probably hundreds of threads on discussing the "best anti-bear handgun", particularly over in the handgun forums. You will find more opinions and info that you could ever hope for.
.45 acp is well below reasonably adequate for trying to stop a brown bear.
Your best defense (other than awareness/precautions/avoidance) is pepper spray. It'll work faster and better than a handgun, by all accounts.
If you run out of pepper spray, and cannot employ your longgun for some reason, you'll want a minimum of .44 magnum with heavy hardcast loads vs. a brown bear.