Are you sure it is not an allergy ? Sometimes we wake up with that, runny nose , itchy eyes and throat etc and it is actually an allergy not a cold. I remembered when my daughter was a baby , she had a blocked nose and had to breath through her mouth . My doctore recommended Dymatapp , which gets rid of allergy symptoms and next minute she breathed well and slept so much better . Check with another doctor and/ or pharmacist and see what they think. With any meds, follow strict instructions for children .

try to keep him warm and comfort foods

I took my daughter (14months) to a pediatrician and after a couple of weeks messing with ideas and meds we have finally agreed on Asthma and the need for a preventer pump and ventolin for bad days. this will hopefully onlybe needed in the colder months. I know for a fact that my 3 yr old could have done with this preventor 2years ago yet no one will diagnose her because shes still young. doctors follow books too often where as pediatricians tend to hunt for the answer and continue trialling new stuff. follow your gut, no one should be coughing all the time especially so young.

If you have the windows open at night could it be an allergy to something in the garden.
I had a similar problem when I had carpet on my bedroom floor. I also had earache with cough etc. I was sent for Chest Xrays which were clear. If it is viral it is taking a long time to clear maybe he needs a suitable cough mixture. If you get nowhere with the GP, perhaps talk to a pharmacist to see what he/she suggests. I got more help from them on a couple of occasions than I did my GP.

I would take him to a doctor again for a second opinion. Chances are the doc is right but at least it will set your mind at ease. Try taping your kids cough on your mobile phone. This way the doc knows what you are dealing with. My son seemed to only have a cold during the day but croup at night but until the doc heard the recording he didnt really believe me. As he is one of the kids that are bright and happy and full of energy even at their sickest.
Suggestions for helping the cough in the morning it to keep the air in the house warm by having a heater or central heating on to take the chill out of the air. Or put some vicks ( either baby vicks or normal depending on his age) on his chest.Thats proberly all i could suggest at this point.

It might be a good idea to get another opinion or request a urine test &/or blood tests for possible diabetes or other things. My caution comes from my own experience with similar symptoms.