The recent book "Tracing Your Naval Ancestors", by Simon Fowler (Pen & Sword, 2011) lists on p46-48 various biographical texts about RN officers, many of which are on-line at www.archive.org and Ancestry. It may be that some of these could provide a clue. Ancestry, e.g. has a database "Commissioned Sea Officers of the Royal Navy, 1660-1815" imaged from the book "The Commissioned Sea Officers of the Royal Navy, 1660-1815" Vol. I-II. 1954[?]. However, this seems to give no more than the basic dates in his ranks (Commander 5 June 1793, Captain 26 Oct 1795, Rear Admiral 4 June 1814 - no date for Lieutenant).
Potentially more useful is a CD from 2003 "The Complete Navy List of the Napoleonic Wars 1793-1815" by Patrick Marione. The description suggest this contains biographical data. There is an index to this online at http://www.ageofnelson.org/NavyList/results.php?name=campbell
Both Donalds are in the index - ominously there is no birth date for yours. The book says that the Society of Genealogists has a copy and other large reference libraries may also have it.
On a different tactic, http://archiver.rootsweb.ancestry.com/th/read/SCT-ISLAY/2009-09/1252735710 shows that someone believes his father was "father: Donald Campbell listed as Major Donald Campbell on his Islay headstone" - I think that's the father was a major. And one possibility would then be to see if ScotlandsPeople has a will for the elder Donald. All getting very speculative but there is a testament for a Donald Campbell, "Colonel of Glensaddel" dated 22/6/1801 and another for a Donald Campbell, "of Glensaddel" (maybe the same person?) dated 1/11/1800. (dd/mm/yyyy format dates)
Unfortunately, the earlier one is a Testament Dative - ScotlandsPeople states
- "A testament dative is comprised of three parts: an introductory
clause, an inventory of the deceased’s possessions, and a
confirmation clause. There is no will, but the court often named a
close family member as executor. However, if the deceased died in
debt, the court might appoint a creditor as executor instead. In the
latter case, it is less likely that family names will be included".
The 2nd document is an Eik, which SP describes as "An addition or supplement to a deed".
So at risk of upsetting my ancestor's countrymen by using English terms, the 1st is an admon with inventory, not a will per se, the 2nd an amendment to the 1st. Whether any of these will mention the Admiral is totally unknown. Depends how desperate you're getting!