D.B.
I had chronic bronchitis for many years - like your son, I did have weeks when I had no symptoms, but it was a "wet" cough like you describe. It was worse at night due to lying down and having things drip down (post nasal drip) and also most health professionals will tell you bronchial things are worse at night. I have a friend who had terrible allergies, and her son was a lot like yours at 1 year of age. He spent a year being sick for 3 weeks, semi-well for 1 week, then sick again. This went on for a year. Interestingly, all 3 of us have become virtually symptom-free (and almost entirely medication-free) by boosting our immune systems without medication.
And I think the doctor is right not to force your child through a lot of invasive tests - your only option would be medication anyway (including nebulizers and so on), and you really have to question whether all those pharmaceuticals and chemicals are good for your child long-term. Meantime, they are only addressing the symptoms anyway, and not the cause.
The cause is not day care and germs - we are all exposed to germs all the time anyway. The cause is a somehow depressed immune system, which is so common in most of us that the AMA said in 2002 that our diets are insufficient to give us the nutrients we need. In all of my work in the past six years, including training seminars with top pediatric nutrition experts and experts in human genomics (dealing with immunity and other issues), I have heard nothing to contradict using the new discoveries in epigenetics and super-foods to bolster the immune system. This sub-cellular nutrition level has completely surpassed the "macronutrient and balanced diet" philosophy of the 60s and 70s and the "micronutrient & individual vitamins" philosophy of the 80s and 90s and early 2000s. None of those worked, and we are sicker and with more chronic and auto-immune diseases than ever.
I'd pursue that before I'd do anything more medical, and I wouldn't get rid of the cats & dogs. You still have so many environmental influences that you cannot avoid - it's a shame to see people getting rid of pets, pulling up carpets, cutting down flowering trees, and going through incredible elimination diets, when it's just not necessary. Maybe a few people have gotten relief, sure, but at what cost? The financial implication, the incredible vigilance about reading labels and being "everything-free" in the diet - and still not getting major relief for most people. It's not productive. And it's not backed by any of the new research over the past 15 years.