Sounds like allergy asthma. My sister used to cough and cough and cough. My parents would walk the floor with her.
It ended up no animals in the house, at all. Her stuffed toys were given away. No carpets in the house and everything on her bed washed in hot water. And allergy shots, twice a week at first, and then once a week for years.
The upshot? Normalcy as a teen and an adult. She still can't get around a cat without her eyes getting red and her nose running and sneezing, but she doesn't end up coughing for hours on end and ending up in an oxygen tent in the hospital. (That started happening the 2nd year into this coughing thing...)
You need to go to a pediatric allergist again and have a full panel of testing done, including blood work.
Good luck,
Dawn