There is a probiotic called Benebac that is very helpful after antibiotics, and it is not very costly at all. Personally, I'd give some after any antibiotic, but especially after a strong one like doxycycline. The only thing that occurrs to me is that my daughter and I do get the kind of cough you describe after we are sick. Viruses and infections just hit us hard and last longer than they do with other people. I run air purifiers where we sleep and vacuum and dust often. It makes a huge difference for us. also, I have an old crock pot near my bed, and I heat water at night to humidify the air. This makes an enormous difference in the cough and recovery time.
Maybe you're already doing things like this. If so, please disregard.
Also, I am not saying you have dry or dusty air, but rather that whatever is happening may be worsened by the air.
One thing I have found with any creature we have had, is that sometimes you have to wait for the illness either to show itself or go away. Not sure if you are comfortable doing that, but when testing options run out, that's what I do for us and for the fids, too.
Please keep posting, since this is how we all learn. I appreciate the exchange of information and different approaches.