I'm sorry to hear about your bichon. I have 2 concerns for your dog.
Smaller breeds like yours can have a tendency to have a weak trachea, often referred to as a collapsing trachea, which is essentially what is happening to it. The trachea is made of cartilage rings and over time in smaller breeds some of the rings get weaker and more closed up than others, thus causing a cough from the irritation. Rarely does it affect a dog's activity level, attitude or behavior, unless it becomes a more severe collapsed trachea over time, which then results in difficulty breathing as well as surgical fixing. Most dogs do not get to this point, but we do suggest, if not already, not using a collar/leash on your dog but instead a harness to lessen any tension on the trachea. Your vet can palpate the trachea safely and initiate this coughing by doing so in most cases if this is the problem.
Another option to consider because your dog is older is a problem within the heart. Heart problems can be troublesome in that serious symptoms are rarely seen before a major problem occurs. While a cough doesn't seem like a big deal, it could often be the first sign that their is a valve problem within the heart. Because the cough is dry, that's a fairly good sign no fluid is built up around the heart that shouldn't be, however, the other side of the heart having difficulties can lead to fluid build up in the abdomen, or even eventually a problem in the chest cavity.
The general rule with ongoing coughing is to have the heart listened to by your vet well to detect any murmurs/ signs of a problem, and if needed, an echocardiogram to determine what the problem actually is. Though heart problems cannot be cured in dogs, medications if appropriate can add a lot of quality and quantity to their lives. Major symptoms first seen with a heart problem include coughing when resting, coughing at night, and wanting to get up and off their chest when the coughing starts up, due to the discomfort in the chest cavity.
Because we don't know which problem is occurring, OTC medications are not safe. Trying to help a cough with a collapsing trachea without knowing this is what is happening can potentiate a worsened heart problem.
Hope this helps, but please don't hesitate to reply if I can help further with this. Good luck with your dog,