My daughter is the same!! She just cant seem to like having the blanket over her and will push it away as soon as I cover her with it and even when shes asleep she knows to kick it off (amazing i might add).
When the weather gets cold I find skivvys, tights and bodysuits are the best to keep her warm. I would wear all this for her and if i feel that shes stil not warm enough I add on flanelette pjs.
But my advise for you is to trust your instincts and dress him accordingly but keep in mind that at his age he is strong enough to regulate his body heat to keep him warm so he only needs a extra layer more than what u wear (more if u use the blankets to keep warm, considering he doesnt want the blanket).
Other factors may be the season, not all coughing is because of the cold weather. He might be allergy to something in the air. Cold weather alot of ppl cough in winter or spring and it might be the pollen in the air.
I know we all love our little ones but try not to stress too much.
Good luck and all the best,