If he's only hacking and coughing on the leash then its probably just from pulling. If he's coughing off leash I'd be more concerned about disease.
Has he had shots for kennel cough? Even if he doesn't have kennel cough now I'd recommend getting shot if you frequent parks where there are other dogs. Sounds like you definitely need a better vet, you shouldn't have a vet that makes you not want to bring your dog in.