Our daughter also had a "non-progressive cough". I could hear every cough every night for years.
It was finally diagnosed as a coughing asthma.. But this took a long while until they would call it this.. So we were like you.. trying everything.
We live in an area with lots of dust and pollens and so we decided to make our home, especially her room a breathing paradise for her.
We pulled out all of the carpets. We got rid of all of the drapes and made sure we washed her bedding more often.. I would purchase 2 sets of bedding so that one comforter could be washed over and over.. and the other was saved for special occasions. .
We also purchased a good filter for our air and heating system .. The best thing we did was purchase a Hunter electric filter for her room. The literature says to change the filters every 6 to 8 weeks, we discovered hers needed to be changed every 4 to 6 weeks.. a few times on the 3rd week.. It was covered in dust.. All of this changed her life and ours.
She did have a nebulizer so we could give her treatments and she then went on an inhaler.. Thank goodness she outgrw all of this by 6th grade..