Can you get a referral to a pediatric allergist/pulmonologist at Mayo based upon your location? I'm surprised he recommended alternative/homeopathic solutions for something as serious as asthma.
As much as it stinks to have your child tied to medications, I will say that the products on the market have the monitoring of the FDA behind them. If there is a new precaution, recall, etc. you will know about it. You will not with a homeopathic product/treatment. It doesn't mean there aren't any risks - just that there's careful monitoring and a track record of clinical information that led to the product being approved.
That being said - I have a friend who's been a doctor ~8-10 years and had dealt with asthma all his life. During one particularly bad attack, he was tested for food allergies and found his asthma stemmed from a gluten allergy he had never known about. I'd see as many specialists as possible to make sure you completely understand the root of his issues and how to most safely/effectively treat him.