The pulmonologist is right; whatever the cause, your cough and wheezing should be treated with asthma medicines. He may be right to suggest that this is a viral infection but you should also be checked for bacterial infection, sinusitis and other causes of cough. The cough may be asthma related or maybe not. That should not be assumed. Your pulmonologist should look to diagnose other causes of cough. One diagnosis, often not considered, that is characterized by a terrible cough that can last as long as 3 to 4 months, is pertussis, commonly called whooping cough. One reason it is not considered is that many doctors do not realize how often it occurs in adults. You might want to ask your doctor about this possibility. Also, if it has been quite some time since the chest x-ray, you should have another. And, at the same time have a throat culture.
Good luck.