30.06 is more than enough. Any bullet, 150 gr and up would do the job nicely. Make sure you read up on bullet placement though, cause it can be decieving as to where the boiler room is on a bear.

Ditch the .357. I live in Canada where you can't carry handguns for hunting. No one ever has any issues taking a bear here with "just" a rifle of shotgun.

Take the shot, listen for a death moan (you'll know it when you here it) and go find your bear.