There are many things that can cause chronic coughing. I am impressed with your care for Teddy: X-rays and lab work can be very costly! I am also impressed with your vet to be so thorough.
If the chest x-rays are normal then it probably isn't heart failure. *phew*
Heartworm test negative would rule that out for you.
Lymphocytophilia can cause constriction of the airways, but it is usually rapid and then ceases.
Bordatella or kennel cough is possible and the culture will tell you more about that. Like humans dogs get many upper resp. tract infections that are bacterial and viral. Most are able the "throw" them off by themselves, but sometimes the weakened immunity will cause them to get a secondary infection. If this is the case and the infection is bacterial, then the antibiotics should help.
Because this is a repetative problem i would maybe think allergies- if it is always at the same time of year, but I don't really know.
Give your vet a call and ask if you are really worried. They should be able to tell you what they are thinking it might be, but the culture will hopefully give you your answers.