To simplify things... You're the customer, you're paying for these appointments to be told you're over reacting. Ask for testing, if the doctor refuses, the tell him you'd like that written in the file that parent requested culture, certain tests and he refused because....
Basic fact of the matter is there's something wrong with your child, you'd like answers. As for the broad spectrum abx, I guess I'd ask for a culture to see what's growing in the sputum or mucus sample and see what antibiotics it's sensitive to. If she's culturing pseudomonas and they're using a regular run of the mill abx, it probably won't do any good. Years ago I took our then 2 month old to the doctor with a horrible productive cough and was told by the doctor "they cough, it's what they do" -- meaning people wcf. At the time he was on an antibiotic (Keflex); however, he ended up having bronchitis caused by enterbactor choclae -- a bug common in NICUs and needed two completely different antibiotics to clear up the infection.