S.H.
Have you taken him to the Doctor and had his cough evaluated?
That would be best....
many things can cause a cough...
If he gets any worsening symptoms or fever.... he needs to go to the Doctor.
As an aside: Pertussis is 'epidemic' in some States.
If he can't nap because of the cough, then he probably cannot sleep at night either. So either way, he is getting lack of sleep.... which affects the body's recovery and healing...