"Negotiator" needs to go off on the final turn (before he dies) in order to get the bonus to drop rate. Negotiator stacks with itself, so it's possible to proc it multiple times with a double pincer (2 guys each side) or multiple Negotiators. Skill trigger rates go up to 100% (not drop rate of loot, but Negotiator has a base 30% trigger chance) if their users are chained to a Power Point, the P circle that spawns after some time.
Levi will likely go away within 30 hours, it's unknown if or when she'll be back, so I'd grind Levi and the Co-op boss for the next 2 days.