Any rifle that will kill a bear is bound to be heavy. Honestly in your case I'd pack bear spray and learn how to deal with the occasional bear encounter. Shotguns and the attending ammunition are very heavy. the only shotgun ammo I'd trust on large game is a rifled slug which is just about the heaviest.

I'd try for the handgun permit first and if successful carry what I do when I feel it necessary,and that's rare. What's the worst that could happen? They say no? I have a Ruger Alaskan chambered in .454 Casull. It's a beast to carry and shoot but will dispatch large animals well. However after thinking about it I believe it's been with me once.

With that being said you've survived an encounter w/o a firearm why do you fell the need now? Did it spook you or is this something you've thought about for a long time?