Such a cough can be infuriating and terribly distracting. The information you have provided is consistent with the diagnosis of asthma, as your doctors have suggested, and I suspect it may well be the cause of your cough, despite your disclaimer, “we all know our own bodies and that asthma hay fever and eczema all come under the same umbrella, and I don’t and have never had them”
That you have not had asthma, in the past, does not exclude the possibility that you have it now. Cough variant asthma is a well recognized entity; asthma without wheezing, and cough as the only symptom. The other most common causes include post-nasal drip and Gastroesophageal reflux, but there are dozens of other causes, infectious, inflammatory and allergic. Even occasional chronic cough caused by an undiagnosed foreign body. Further investigation should include testing to rule out all of the preceding.
In your case, the apparently very good response to steroids is encouraging and suggestive of an allergic cause. For that reason and because further evaluation for a thus-far hidden cause is warranted, I disagree with the opinion that your only choices are to “live with it” or take steroids the rest of your life. Further testing to rule out (or in) the diagnosis of asthma is warranted, as is a trial of inhaled steroids.
Bronchoscopy to rule out disease in your larynx, trachea or major airways, disease that might not be evident on a Chest X-ray or even a CT Scan is a reasonable suggestion.
If after the preceding, your cough persists, I suggest that you seek another opinion from a Cough Clinic, at the nearest academic medical center or major clinic such as the Mayo Clinic. Most academic medical centers now have Cough Clinics, to address the very common problem of chronic cough, such as yours.
Good luck
I sense your fustration; while your waiting to have the bronchoscopy, try a combination of Prilosec and Zantac for 2 weeks. Both can be bought without a prescription. Should your cough resolve or not, see a gastroenterologist so they can take a look down your esophagus.