I'm glad to say that we are finishing this set of posts on a highbrow note. I have disqualified Dan Brown's Angels and Demons, not on the grounds of literary quality, but because as far as I remember much less than half of the book is actually set in the Vatican, with other parts of Rome getting a good deal of coverage.
Instead, the top book with a Vatican setting, on both Goodreads (by a wide margin) and LibraryThing (by a smaller margin) is a 2003 work of non-fiction, recounting the story of how one of Europe's greatest artists spent four years creating probably the best known artwork in the Vatican. It sounds excellent and (when I eventually have some time) I will try and get hold of it myself. It is:
Michelangelo and the Pope's Ceiling, by Ross King
There are an awful lot of thrillers with Vatican scenery (looking particularly at the works of Daniel Silva here), and I have spent more time than is really necessary looking for those where the majority of the action is actually set there rather than just visiting to have a look. The most convincing candidate (on that criterion only) is a novel regarding the revelations of Fatima, spilling over into a papal conclave (which let's face is is the only newsworthy thing that ever happens in the Vatican these days). It is:
The Third Secret, by Steve Berry
Appropriate enough to finish up in the Vatican on Good Friday!