I've had bronchitis (which was then called asthma) my entire life and it started when I was a month old and contracted pneumonia. As I have grown up (i'm now 46) I have studied intently myself, my kids, sisters, friends who have chronic bronchitis.
All your baby's symptoms, were just what I felt. You are itchy on the inside of your body before the other symptoms start to show. Then the snott would start streaming, my nasal passages would flare up and feel tight and burning then the throat would close up and my lungs would panic. My parents used to rush me to the hospital and they would pump me with antihistamines (in a needle which made it much worse for me because I was already terrified). Then they kept me high on antihistamines for days afterwards which dried out my nasal passages and my lips would chap and my breath would get foul, but my nose wasn't running.
The long and short of it...I was allergic to everything...We had no cats in the house. It came as a result of pneumonia. Seems to me (and I have gone to lectures and read results of loads of research into asthma) that I developed a bacteria in my lungs as a result of the pneumonia. When you cannot deeply exhale (as when you have phenumonia) the air becomes stagnant and develops an infection. One begets the other. I was not allowed to wail as a baby. My father could not stand the crying so I coddled when I really needed to have a good scream. Screaming cleans out the stale air in the lungs (babies don't get any exercise to generate oxygen other than screaming).
I started deep breathing techniques I learned from a brochure the american cancer society handed out about emphysema and bronchitis/asthma. My mom had phoned me just two weeks prior and told me she heard on the radio in an interview with a doctor who said what perpetuated and asthma attack was not that the person wasn't getting enough air into their lungs (which is what I thought was happening) but that the carbon dioxide wasn't coming out. I grew up with inhalers by the way.
So, when I next had an asthma/bronchitis attack I didn't use my inhaler, I sat down and forced myself to relax and try breathing out more with every breath, and not let myself take in such huge amounts of air when I felt like gasping for air. It was so much work to do that and I was terrified the whole time. But do it I did. I did this for every attack for the next two weeks and then when I received the brochure, as I mentioned above from the acs I started to do the breathing exercises and within one month I not only never had another asthma/bronchitis attack, I actually never had another runny nose or even a cold from that month on. That was 6 years ago.
I realize your child is a child, but having them relax and practice breathing techniques twice a day will clear their lungs and force the bacteria to dry up and go away (make it a game). Antibiotics don't work...My mother was big on getting me to the doctor and that was forever their remedy other than a humidifier and over the counter antihistamines.
It's all about relaxing and breathing out. I had attacks at the weirdest moments too, but I know the methodology now that I look back.