Plot hole: One of Snape's memories show Lily telling baby Harry to be safe, be strong. That had to have occurred before Voldemort killed Lily. Snape was not in the house until after Lily was killed and Voldemort was gone. Only Lily and Harry should have had that memory, but not Snape.
Plot hole: Immediately after the confrontation between Harry/McGonagall and Snape in the Great Hall, Harry runs up to Ravenclaw tower against crowds of students running the opposite direction. However, all the students would have been in the Great Hall during that confrontation, and therefore would have been coming from the same direction as Harry.
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.