I have had SLE for over 30 years. but I have only in the last couple of months started to have really prolonged problems caused by coughing. I don't think it has anything to do with my having lupus though. I also have Sjogren's and have tremendous problems with dry mouth, amongst other things. The dryness has extended down into my throat and my gullet, and I honestly think this might be the cause of my coughing. I use an artificial saliva spray which does help with the dryness of the mouth and so helps with the throat dryness, and this helps with the coughing. Maybe you could ask you doctor about this.
I also have asthma - late onset - which is not allergy related. The thing is if you do use the sprays, they have a tendency to make your throat dry which given how you are at the moment isn't the best thing for you to have happen. The best thing to do is to use a spacer to inhale them, and after using them rinse your mouth out. This will help to lesses the dryness that this can cause as well as get rid of the horrid taste.
I have never noticed in all of the years since diagnosis of SLE and of taking steroids daily since then that I have ever had a cough as I have now, this is why I am convinced that mine is caused by Sjogren's. When I am unwell, I increase my steroids myself and this does not help with the cough in any way.
Obviously this is only my own experience, but I hope it will give you some ideas to work with and that things improve for you soon.