This is obviously a very complicated case. Seeing the Pulmonologist is an excellent idea. The cough could be from reflux, possibly a sinus infection, asthma, or even from a persisting pneumonia. It is difficult to tell with the little information you have described. A good physical examination and possibly some x-rays should be able to clarify the source of the problem and lead to correct therapy.
could the cough be from excess mucous from the sinuses or past nasal area? I had a persistant cough for years that never got solved until I eralised it was due to food allergies for me - gluten and dairy give me tons of mucous and coughing fits. I suppose the child is on solids and may be reacting to wheat and dairy. Did the diet change around the time the cough started? Just a thought. Best of luck.
HI,MY 10 YR OLD DAUGHTER HAD CHRONIC COUGH FOR OVER 5 MNTHS-DOCTORS HERE KEPT TREATING FOR ASTHMA AND WOULD GIVE HER PREDISONE(STEROIDS)AND INHALERS-FINALLY GOT SO BAD SHE BEGIN TO SPIT BLOOD AND BLOOD IN STOOL,GOT FED UP AND TOOK HER TO LEBONHEUR CHILDRENS HOSPITAL IN MEMPHIS 3 HOURS AWAY,THEY TOOK HER RIGHT IN AND BEGAN A SERIES OF TESTS FOR CYCTIC FIBROSIS,TB,WHOOPING COUGH,THEY DID NOT STOP UNTIL THEY FOUND THE ANSWER,THEY SCOPED HER LUNGS,UPPER AND LOWER GI,SHE HAD A PULMINARY SPECILIST A GI SPECILIST,AN ENT,AN INFECTIOUS DISEASE DR,AND A IMMUOLOGIST,ALL TOGETHER WE SPENT OVER 2 WEEKS THERE,WELL SHE HAD SEVERE GERD(ACID REFLUX)WHICH AS YOU KNOW CAN CAUSE A BAD COUGH,SEVERE SINUS DISEASE WITH 100 PERCENT BLOCKAGE,AN INFECTED ESOPHAGUS,AND THE THING ABOUT IT SHE DIDNT EVEN HAVE ASTHMA AND THOSE MEDS SHE WAS ON ESPECIAALY STEROIDS DONE HER BODY HARM AND MADE HER GAIN OVER 20 LBS IN THESE MNTHS,THE STEROIDS MORE LESS MESSED HER ESOPHAGUS UP THEEY TOLD ME,SHE HAD SINUS SURGERY A FEW WEEKS AGO AND GOT EVERYTHING ELSE UNDER CONTROLL AND DOING ALOT BETTER-SO PLEASE DONT LET THEM KEEP PUMPING PREDISONE INTO THIS SMALL CHILD UNLESS DIAGNOSED FROM A SPECILIST-HOPE YOU FIND YOUR ANSWERS-LIITLEVICK0