I Have Had A Chronic Cough X 3.5 Months. I

I have had a chronic cough x 3.5 months. I have had 2 CT scans (both without contrast) and a breathing function test. My current working diagnosis is eosinophilic bronchitis. I am on a daily inhaler (Dulera), Singulair, and Zyrtec. When I add Prednisone (20 mg / day) my symptoms subside. After 2 weeks at that dose, I was told to wean down to 10 mg per day. My cough (productive, with chunks of thick, sticky yellow sputum) has returned. I also have sleep apnea and use a CPAP every night. I have had a recent sleep study so I am confident the current setting for my machine is accurate.
The next available appointment with my pulmonologist is not for 2 weeks. Do you have any thoughts on what we may be missing? What tests I should ask my doctor about doing? Other therapies to try?