Revealing mistake: In the scene where Harry and co. are in the Room of Requirement, at several points during this scene, you can tell by the way the light shines that Harry's glasses are missing their lenses, especially in close up shots of Harry.
Revealing mistake: When Lupin is on the castle bridge and reunites with his wife-right after the scene where Neville beckons the dark wizards that disintegrate in the force-field-if you carefully watch the right hand corner of the screen, a very obvious green screen error can be seen in the castle window.
Revealing mistake: When Gregory falls in the fire and dies, while he is falling, you can very obviously tell that it's a stunt double who looks nothing like Josh Herdman.
Chosen answer: Hermione knows this is an unusual request and she and (and Harry) suspect the bank knows that Bellatrix's wand was captured by Harry Potter (at the end of Part 1). Showing the bank manager the real wand would prove that this is not Bellatrix. The bank and Voldemort's minions expected that Harry and the others might go to Gringotts searching for a Horcrux in the Lestrange's vault. They are setting a trap for Harry and his accomplices.
raywest ★
Incidentally, if I remember right, in the book Hermione actually does hand over Bellatrix's wand, not realizing the Gringotts goblins already know the wand is no longer in her possession, forcing Harry to cast a timely Imperius Curse to avoid the entire plan being exposed.