A few possibilities here, but what you are describing sounds like it may be tracheal collapse. This can happen in toy and small breed dogs as they get older, and it is characterized by a dry, honking cough that is usually worsened by exercise, obesity, heat, humidity, and exercise. The fact that he improved when he lost weight supports this possibility.
You may want to look up "tracheal collapse cough" or "collapsing trachea cough" on Youtube to see if it appears to be similar.
There are a couple treatment options that you might want to discuss with your veterinarian, including cough suppressants (hydrocodone, butorphanol, guaifenesin), anti-inflammatories (prednisone), and bronchodilators (theophylline, terbutaline).
In an older dog, your veterinarian may want to do a couple things to rule out other underlying problems (mass pushing on the windpipe, lung disease, etc) before trying medications, but with the concern over sedating him to take a look, it might be worth trying the medications first to see if they help.
Hope that helps. Please let me know if you have any other questions.