Oh, we need more nurses in this country..such a shortage..thats great you are one!
And you brush his teeth..kudos to you, many people dont do that!
A dry cough like that could mean many different things, so of course I can't tell you what it is, but can give some thoughts for you to consider.
As you indicated, it could be an allergy-related cough. This is a llittle unusual in dogs, as their allergies are mostly skin-related and dont bother the respiratory tract, but they can get allergy-related coughs.
We always need to be concerned in a dog of that age of other problems that can occur.
1. Chronic bronchitis does happen and can present as a chronic dry cough.
2. Heart disease
, including that due to heartworm, but also non heartworm disease. In dogs as they age they can get other disease of the heart.
3. We would have to consider a mass of some kind that could be pressing on his trachea inside his thorax.
At this point I would suggest a trial with benadryl for allergies. (Generic is ok too) Typically it is used at 1mg/lb every 8 hours. You just need to get the dose close based on the pill size availablity, it does not need to be exact. It is very safe but may cause some drowsiness.
However, if he seems to be getting worse, and this does not provide some improvement within 10 days, then I would suggest he go to your vet for a checkup to see if they can sort out what is going on.
I hope this answers your concerns. If you have any more questions on this, please let me know!
As always, feedback is appreciated!