I would absolutely not put peroxide in a nebulizer. That could be harsh on the lungs, and could be a problem if it got in the eyes.
If there is really an infection issue, I can tell you what has worked for me. I've used Colloidal Silver in a nebulizer with no ill effects. I have found it to be as effective as Antibiotic for lung and sinus infections for me. When I have any infection, I also take Colloidal Silver orally, as well as various herbs. I'm not sure about safety for children with the herbs though, you would have to check. Here are some good resources for using Colloidal Silver in the nebulizer, and for what might be helpful to take to fight an infection.