The breeder shipped out seventy-three puppies?
It sounds like you either dealt with a broker or a puppy mill. No wonder the little guy is sick. I hope you get reimbursed for your vet bills.
If both vets didn't seem to think there was anything to worry about and he's eating and playing, I'd just wait out the three weeks and see if he stops coughing. I know it's stressful being a new mom, but it sounds like your vet has it under control.