It's important to separate possibilities from probabilities. I'm slowly learning this. Is it 'possible' for a 747 to come through your window at 3:45 am? Well, yes, there is a way for that to physically happen. Is it 'probable', no.
In the middle of a rabies scare myself. Coping better this time, so yay. Anyway, a few facts I've learned:
* rabies is managed by erring far on the side of caution. Since it is untreatable once symptoms start, vaccine is often given in 1/1000000 type cases because no one wants to get sued.
* rabies shots for dogs are scheduled every 3 years, after the 'puppy' shot that's good for one year. The dog was only a year or so overdue at the worst case.
* rabies shots are effective far longer than the scheduled interval. Again, we see the abundance of caution in that recommendation. Feral cats are managed reasonably well by one or two shots in their lifetime.
* humans are only mildly susceptible to rabies. Even from a known bite, without treatment, the odds are far in your favor. We treat bites aggressively because to wait is too late.
* while the incubation time (between bite and symptoms) can be quite long (weeks to months), the "shedding" interval is very short. Dogs, cats, and ferrets only produce virus particles in their saliva a matter of days before their death. This is why those species can be observed for 10 days after a bite and are no longer routinely euthanized and tested, because if they were transmitting rabies on the day of the bite, they'll be dead or dying in half that time.
So, based on your description, your son got a tiny nip from a dog that still had significant immunity to rabies. That's possible to transmit disease, but highly improbable. Going by the book, if you'd reported it, public health would have not given vaccine, but observed the dog for 10 days. Ten days have elapsed from the bite. So, how's the dog? Is the dog happy and healthy? Then it's, and I hate this word too, impossible that he was infective back in June, and your son is fine. If the dog is dead, then please do call your doctor. Time is on your side still.