Technically quest alone isn't bugged. But after completing it and re-entering Lifter's Refuge, Tiny Woody reappears and you can repeat event in Captain Tatzatl house. You can repeat it infinitely in peaceful way (but you won't get any additional loot) until you kill them (you can get second necklace then).
@Edit: Post was moved here from my thread, but it's not Agentile problem (probably his problem doesn't exist as Cwealm mentioned).