Before climbing out of the window, Bunty removes her scarf/belt and hands it to Mrs. McCarthy.
When she enters the house behind the Chief Inspectors she is not wearing it.
By the time she reaches the library she is wearing it again.
Scotland Yard send Chief Inspector Sullivan to investigate the arrest of Chief Inspector Valentine. A more senior officer would have been given such a task.
When Father Brown picks the lock on the door to the secret library, as the door opens, you can see that there isn't any locking mechanism in the door. The 'keyhole' has been attached merely for the scene.