The Judge Bear (King's Hot Springs Hotel)
After freeing the King's Hot Springs Hotel in Faith's region, talk to the starred NPC for this mission.
The bad-bliss bear is in Lydia's Cave, to the southeast.
An LMG works well here. Like Faith herself, the bear can spawn illusory duplicates. These phantoms can be dispelled by a single bullet. If you shoot a bear and it doesn't disappear, hold down the trigger.
You'll go through this cycle a couple times, but keep pouring lead into the non-ghost bear to complete the mission.
You can loot the bear, but you'll just get regular Grizzly Bear skin.