I'm leaning towards it being fixed at the time of response [courtesy of the lasting effect ruling], which means that Bilbo could get the boost even if Argalad not only reduces the enemy's threat, but if he actually kills the enemy in question. But since that would work out well for the players, perhaps I'm optimistic on that point.
What about picking an immune enemy for the willpower boost? It's clearly "choosing" an enemy, but it does not affect the enemy in any way. Cards that boost based on being lower than engagement cost "interact" with a specific immune enemy, but work just fine -- why should it be OK to interact with the engagement cost but not OK to interact with the threat? (The direct damage wouldn't work, of course.)