Background
I have bronchitis (both asthmatic and allergic). I'm allergic to pollen (among other things) and live in a place surrounded by lots of trees, plants and flowers. I love being outside: working around the yard, exploring nearby trails with my dog, etc..
During this time of the year I can smell the Acacias (I'm specially allergic to those) and the pollution from the passing trucks. I'm starting to get bronchitis just by walking the dog for 30 minutes in the morning.
Problem
I used to practice Muay Thai, but I can't any more (health related issues), so I haven't done any real physical exercise in 5 years. I need to start getting fit again, because I'm getting old and fat, and I want to make sure my health stays in good shape.
I tried running, but since I haven't done it for so long, the bronchitis starts in the first 5-10 minutes. My legs are fine, my heart rate is fine, my lungs are not.
Question
How can I get fit and improve my cardio (lungs)? The easiest way would probably be running, but it's extremely difficult for me. I don't want to get buffed, I just want to make some exercise.
Update
Just to clear something, because I have both allergic and asthmatic bronchitis, my problem isn't only the pollen allergy. The asthmatic part makes running/cycling/whatever a pain. In short, after 5 to 10 minutes of running I already have difficulty breathing because my lungs don't "open" as they should.
Another update
Answering to some questions placed in the comments: I have suffered this condition since I was a baby, from what I know. I never smoked (I'm actually allergic to tobacco smoke). I don't live in high altitude, but it's a very humid area, and my condition got worse when I moved here. Not sure if I feel better when away, will have to wait for the next holidays. My wife has no difficulty breathing.
Yesterday my motorbike didn't want to start, so I ended up pushing it uphill for 50 meters to the garage. Had a bronchitis crisis, so I had to sit down for a while to calm down. I have one inhaler but try as hard as I can not to use it because it causes tachycardia. Actually, riding my motorbike is good for my bronchitis, I believe it's because of the air flow.