Royal Chaplains are part of the Ecclesiastical Household and part of the Royal Household of the Queen. There are two separate households.. one in Scotland and one in England.
Their duty is to serve the spiritual needs of their sovereign wherever they are called upon to do so, but they generally have some other ministerial position outside the Royal Household that pays a salary, since being a Royal Chaplain is not a paid position within the Court.
The Church of England comprises the College of Chaplains, and the associated Chapel Royal, the Royal Almonry Office, various Domestic Chaplains and service Chaplains.
Queen Elizabeth II has 36 Anglican chaplains, in addition to chaplains extraordinary and honorary chaplains appointed to minister to her.
Your Reverend Dunn could have been any one of these 36 Chaplains, or could have been an honorary chaplain or chaplain extraordinary.
So far, I have not found a list of appointments for 1997, but I will post it here if I find it.. However, if you know the area where Reverend Dunn ministered, you may wish to contact the Diocese he was closest to in America. They should be able to answer your question, as they would know if he had been appointed a Royal Chaplain.
I have linked the addresses and contact information for the American Anglican Churches below:
If you know specifically the church he served, you can click on "Parishes" at the above link and it will give you the address and contact information of specific churches by state, where you may get someone who actually knew him and could give you the information you're looking for.
Incidentally, if this gentleman is the same Kevin S. Dunn that was born on 3 November 1931 and died on 11 March 2005, his last residence was listed as Alhambra, Los Angeles Co. CA, according to Social Security records. And the closest Anglican church to that location appears to be St. Luke's Anglican Church in Glendale, CA.
Hope this helps you in your research.