Chronic cough can be a sign of asthma or as your previous posts mentioned "cough-variant asthma". Sometimes it's the only symptom of asthma. Don't get nervous by the word "asthma". This can go away or can just be a symptom at this time that he is exhibiting.
Do you have a nebulizer? Very important to have one on hand. Keep in mind - the bronchodilator medications that are used in the nebulizer can also make a child more energetic/aggressive. Someone mentioned flovent and, those medications are cortico-steroids. They are used to reduce inflammation and are are a good preventative medication, however, they can also pose side effects. If this is an asthma-induced cough and is so severe that his face turns red, he should have some kind of bronchodilator/nebulizer on hand. I've had asthma for over 20 years and am very familiar with these medications. I have even gone through allergy shots for many years.
If your child is off any kind of antihistamine for at least 48 hours, your doctor can write a scrip to bring to a hospital to get the RAST test, which is a very simple blood test that can give you an amazing amount of information, in days. You don't need a specialist to write a script - request it from your pediatrician or ENT. They are giving you the run-around! He could even have a food allergy which is something to discuss with the drs. Yes, GERD can be a concern but with his history of RSV this past winter, I would still be focusing on the bronchials/lungs/breathing. Regarding medications, unfortunately, for children, the side effects can be much more noticeable or they can be more sensitive to them.
Someone mentioned "whooping cough"....I believe your doctor would have been able to pick up on that. My son, at a very young age had RSV and it took a long time to rid him of the cough (he took albuterol-a bronchodilator-in a liquid form for a short time).. I, too, questioned whooping cough, as it has started to make a comeback. The dr. heard his cough and ruled it out. Hopefully your dr. has done the same and eased your mind.
My children have taken Claritin without any side effects, yet, we've tried Zyrtec and Benadryal and both have side effects for us.
Yes, his aggression could be age but with the symptoms that you are describing, he sounds like he has an underlying medical concern and it's good you are addressing it. At the very least, get a script for a nebulizer and medication and see if the dr. will give you that medication for the RAST testing.
Good luck.