Is It Appropriate To Start My Toddler On A Year-long Budecort Inhaler For Allergic Cough Without A Chest X-ray?

Hi my baby is 4 , 19 kg. She has been having cough for almost a month but it increases n reduces in intensity .. it has again intensified. I had been giving her asthalin 5 ml for 7 days, maxtra syrup 4 ml also for same days then there’s still not much improvement .. I went back to the doctor n after checking with his stethoscope he concluded it’s allergic cough n she will need inhaler for a year with budecort200 mdi .. when I asked why for a year!! He says it’sa preventive as she’s prone to congestion from allergens.. Then told me to start her on amoxicillin syrup 5 ml x3 for 5 days and asthalin 5ml x3 5 days.. I’m totally confused n scared if he’s treating her right as he never took an X-ray of her lungs before telling she needs inhaler for a year… my son has the same coughing issue he’s9 yrs.