Probably your son is suffering from allergic bronchitis
that's why your doctor is prescribing budecort
, you not mention than whether is giving in nebulisation or oral, if it giving in nebulisation form then it can be givien for long time, not much harmful.
Regarding antibiotic i must be clear to you that, antibiotics are not to treat cough
, their usefulness only to treat infection whether it is associated with cough or not.