If it only happens at night, maybe it's something in his room that he's allergic to. Dust mites, the detergent you use on the laundry (it may not bother him as much on his clothes, but close to his face on his pillow might), a pet that sleeps with him, a plant in his room or outside his window, the bath soap he uses right before bed, a snack he has before bed (could indicate a food allergy) or anything else that may be specific to bedtime. Rather than just treating the symptoms, it's probably more important in this case to find the cause. It also might be easier to find, since it only happens at a specific time. Ask your child's doctor about whether giving him Zyrtec every night is okay or not. Also, make sure you use an actual teaspoon, not a kitchen one. The kitchen ones are bigger and you may end up giving him more than 1/3 more medication than he should have.
Have his lungs been checked for lung issues like pneumonia? It's unlikely that it would go this long, but since it started abruptly and is pretty severe, it wouldn't hurt to ask about it.
The fact that it happens at night (when he's presumably laying down) may indicate acid reflux. The acid irritates the throat and causes coughing; sometimes it can even be breathed into the lungs and cause asthma-like symptoms and even pneumonia. You may want to ask the doctor about that and see if you should try acid reducers or if he should be tested for acid reflux first. Don't give him the medication without asking the doctor first.
He could also have some kind of sleep disorder, like maybe sleep apnea that causes his throat to relax and close while he's sleeping. Often, sufferers wake up coughing and gasping for air. If the asthma/allergies and acid reflux are ruled out, I would take him to a sleep clinic or sleep specialist and get him checked. Once it's diagnosed, sleep apnea is easily treated with either a dental appliance for mild to moderate sleep apnea or a CPAP machine for more serious cases.
Please write back if you get more information from the doctor or if you have any more questions. Best of luck to you and your son.