Both the staff car that Carter rides in (as Hitler) and the vehicle that Burkhalter comes in have the same license number - WH 54693.
In the barracks, Carter gives an impersonation of Bette Davis. The line he uses, "What a dump", comes from 'Beyond the Forest', a film released in 1949, four years after the end of World War II.
When Carter imitates Hitler, he parts his hair on the left. Hitler parted his hair on the right.
In the barracks, though Carter is not married, he is clearly wearing a wedding ring.
When impersonating Hitler, Carter has a big mole on his right cheek. Hitler does not have this mole. This should have given Carter away. Update: The bump on Carter's right cheek (identified as a mole) could also be a simple pimple.