Your humble opinion would be wrong. This is pretty clearly a "derivative work", which requires permission from the rightsholder.

For example, I could totally write a story that had characters named Gandalf and Frodo. But if my Gandalf happened to be a grey-wearing wizard, who walked around saying things like, "A wizard is never late. Nor is he early. He arrives precisely when he means to," and my Frodo a hairy-footed fellow of diminutive stature with an extra-special ring, I'd better have had permission from the Tolkien estate, or a lot of money to burn on lawyers and fines.

EDIT: Pre-emptively, yeah yeah yeah, parody. Bringing that up in this context is probably diversionary pedantry at best, however.

Bad example. The items in the movie are mostly paraphrases or summaries of the original. This would be more like releasing a movie where in one of the scenes a wall in the background contained a slide that said:

  WIZARDS
  - NEVER LATE OR EARLY
  - PRECISELY ON TIME

Does that require the Tolkien estate to grant permission?

The other factor is that they're background slides.

The Tolkien Trust would be very interested in stopping you from selling a derivative version of Christopher Tolkien's maps but rather less interested if derivative version of Tolkien's maps appear as props in a movie scene involving a convention for fantasy enthusiasts in a movie whose theme and plotline is not even remotely related to the subject of hobbits or Middle Earth.

That's not how fair use works. The only way to determine whether something qualifies as "fair use" is to take the case to court and have a court rule on it.

I'm pretty sure that the documentary aspect of the movie makes the thing fair use.

"documentary"? The Fifth Estate is not a documentary.