Other mistake: In both the books and the movies, Harry's Invisibility Cloak can always be seen until it is worn. In Diagon Alley, just before the Gringott's break-in, Ron pulls out the cloak to cover Harry and Griphook, but it turns invisible before he puts it on them. Ron merely mimes the motion of covering them.
Other mistake: In Snape's memories, Trelawny's prophecy is the wrong one. If you listen closely you can hear her saying "master shall be reunited once more". That's the prophecy that she told Harry in PoA, not the first prophecy Snape half-overheard Trelawny telling Dumbledore.
Other mistake: In the final scene when Harry and Voldemort "connect" their wands, there is a close-up of Harry's face. Although the only light source seems to be the ball of fire between him and Voldemort, Harry's face is lit from his left.
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.