If you allow an NPC to pick up a distraction object then moments before they go into the animation to pick it up, and as they're standing next to the object, you lose the prompt. This is quite normal. But if you nudge them in the moments that follow it breaks their animation, they stand up (and usually comment on how rude and obnoxious you are), and the prompt to pick it up returns.
So if an object's prompts have gone then wait for them to enter the animation to pick it up and you can nudge them to restore the prompt. It's not ideal if you're working against the clock, but it's a reliable way to get the item back.
I suspect this is the reason we're suffering with the problem. The NPC has entered the stages of pre-animation (without actually going to pick it up), so by using it again the item is effectively about to be picked up. Until that animation begins it will remain un-retrievable and promptless.