If you're a handloader there's the .338-06 with bullet weights from 180 to 250 (or more) and it can be built on fairly light .30-06 actions like the M-70 and M-'98 mauser as well as a lot more. It's extremely powerful with over 3,700 ft-lb energy and can send a light weight 180 grainer at speeds fast enough to make the 300 yard range pretty flat. (3,000'/sec)
The best handgun for brown bear is one with a short enough barrel to get into YOUR mouth fast enough to pull the trigger before you make the bear any madder by shooting it with a handgun!
As yet another Alaskan, I offer my 2 cents worth. I'm in agreement with the prevailing sentiments of my fellow Alaskans, the 338 Win Mag will get the job done well. However, the 30/06 has, and will, continue to do it all as well. You'd be surprised at how many life-long Alaskans still use it...and often even smaller calibers. I've got a good buddy who took an 8 foot brownie last fall with his "trusty" 270. Not me though. I have two favorites: No surprises...the 338 Win Mag & 375 H&H and carry a 454 Casull handgun. If you are "at home" with the '06, then bring it on. The only thing I would not recommend the '06 for here is our big coastal brown bears (which we have in abundance here where I live). The brownies require bone-crushing energy and bullets. For them, a minimum of 338 Win Mag. Whatever is used, good shot placement is of the essence.