It is clear that the Inspector uses this chance to get his own back at David's father, having been insulted by him in public previously. In this dystopian world, any new baby can only be openly acknowledged once it has been checked by an Inspector and a certificate signed to declare that it is not a mutant. This had to be done as soon as possible, and so when David's father sends out a message for the Inspector, he expects him to turn up straight away. However, what happens is the opposite:
...the stablehand, instead of bringing back the inspector forthwith, as was to be expected when a man of my father's position and influence was concerned, returned with a polite message that the inspector would certainly do his best to find time to pay a call in the course of the day.
When the Inspector finally arrives, he ambles along in no particular hurry, and takes his time examining the baby, even going as far as to delay at the crucial moment when he signs his name on the certificate. As David comments, it was a very foolish thing to do to argue with the Inspector, as he has "too many ways of hitting back." Crucially delaying in arriving to check the new arrival in the Storm household is just one of those ways, and it is clear the Inspector delights in having power over Joseph after being insulted by him.