If you want a hand gun, I would take the 357 you already have. Even if I was buying a gun for bear I would take a double action, 4 inch barreled 357 revolver over a 10mm. Load it with large meplat hard cast bullets on the slow and heavy side.
There are no brown bears on POW. Hunting on the backside of Revillagigedo I had a mid sized (about 250 pounds) black bear come within 15 feet. It was just curious as we were in some think alders on an overgrown logging road and down wind. Yelling did nothing, but as soon as I started closing distance he ran off. I have a friend that called a solo goat hunt early because two black bear bores started stalking him. Almost all black bear attacks are predatory and can be avoided by paying attention, they are also incredibly rare. Browns are a different story, and browns are what most people think of when thinking of bears in Alaska. I would be comfortable on POW or Revillagigedo without a handgun for bears, I worry about two legged critters more.
