After Batman's cowl is turned bright pink, the footage (stock footage) shows the Batmobile heading back to the Batcave driven by a non-pink cowled Batman.
When the fight starts at the party, Mad Hatter is standing in a doorway filled with people. But when he tries to hit Batman with the bottle, there is nobody else in the doorway with him.
The Mad Hatter claims to have stolen a Foot Guard's shako during Trooping the Colour. However, Guardsmen wear bearskins, not shakos, and the headdress the Hatter is holding is a bearskin, not a shako.
The Mad Hatter pronounces incognito incorrectly. The correct pronunciation is in-cog-nee-toe.