my son started this at 3 months. He had bronchiolitis which is asthma related. I had to take him in every week or 2 for a nebulizer treatment. I don't know why the dr didn't prescribe one for him at home but we didn't get a nebulizer until he was 8 or 9 years old and that doctor couldn't believe he didn't have one already. He had that cough for over a year when he was a baby, then when he got older seemed to have wheezing and coughing in the fall and spring. Now he is 18 and hasn't had any asthma issues for about 6 years.
ok after reading your follow up.. I would find another doctor. Saying that because one breathing treatment didn't do much isn't much help. Sometimes it takes a day or 2 to notice a difference. Sometimes you notice right away. Suppressant cough syrups are not good so don't give any cough syrups. Surprised they didn't tell you to use zyrtec. benydryl is good for runny noses and helps keep the congestion down. If you do decide to try benedryl don't give more than 1/4 tsp.
If he is rattling in his chest he needs to go to the doctor. If your current doctor doesn't do anything then see another one and try to find one that specializes in asthma. Dr Dattel at KC Pediatrics is a great doctor that seems to specialize in asthma. He was the first one that had my kids do a breathing test at one of their regular physicals when I mentioned that they cough during typical allergy seasons.
If he is coughing and rattling to the point of having a hard time breathing, don't hesitate to go to the ER even. An asthma attack isn't something to mess with and is very scary. Luckily my kids had mild cases but I still had to go to the ER a few times with oldest when he was really wheezing and having a hard time breathing.