This is obviously a very frustrating situation for you and your family. It is not clear from your note how the diagnosis of asthma has been made. Has he wheezed or had breathing problems that responded to medicine? Some 5-year-old children are very good about performing lung function tests. Others simply cannot do them.
One of the most important aspects of defining children who have psychogenic causes to symptoms such as these is what happens after they fall asleep. Those children who have psychogenic reasons for clearing their throats and/or coughing often cough up to the point of falling asleep. Once they are asleep these symptoms stop unless they wake up during the night. You might observe your son late at night, at least 2 hours after he has fallen asleep. Look at his breathing and listen to how he breathes. You can learn a lot from that. If there is persistent drainage, he will have some problems even after being asleep.
We, at National Jewish Medical and Research Center, often see children with exactly the same problems you have described. It sometimes takes a few different types of specialists (ENT, allergy and/or pulmonary) to make the diagnosis and create a treatment plan that works. That is one of the advantages of having a center that only focuses on these types of problems. If you are continuing to be frustrated, you may consider a short visit to Denver to sort out these problems and create a way to understand and deal with them at home.