When a straightforward search for a record comes up empty, I think it's important to step back and look at the larger picture, so that one can understand the context of the record -- that often leads to clues about where one might find it.
The question as posed actually has two separate questions in it. One is to find out more about the incident in which Captain Kenneth V. Carlsen lost his life, in order to find out more about him. This may not be possible if the records are still classified, but here are some things to look at that might help.
Q1. USGenWeb publication: a cemetery inventory for Anah Cemetery, Washburn Co., WI. includes the following transcriptions (Transcribed by Priscilla Arnold, Jenny Stilwell and Susie Stilwell in 2002.):
Kenneth V. Carlsen Capt US Army Air Corps World War II Feb 14, 1917 -
Dec 14, 1942 (plaque: Washburn County US Veteran)
Carlsen, Conrad N. 1885 - 1971 Laura A. 1889 - 1982
"In grateful memory of Captain kenneth V. Carlsen, A.C. 0392224 who
died in the service of his country in the Latin America Area, 14
December 1942. He stands in the unbroken line of patriots who have
dared to die that freedom might live and grow and increase its
blessings. Freedom lives and through it, he lives - in a way that
humbles the undertakings of most men." signature: Franklin D.
Roosevelt, President of the United States of America.
The same transcription was also published on this Washburn Co, WI GenWeb site. A similar entry appears in the book Cemeteries of Washburn County and the surrounding area, Volume 1 published by the Washburn County Genealogical Society in 2000 (the link is for the entry on Google Books).
Photos of these memorials can be seen on Find A Grave memorial #33593896, where the contributor of the photos says that the plaque is attached to his parents' marker. [Note that the memorial was created in 2009 and the photos were added in 2011 by a different contributor.]
The plaque says "A.C." which suggests to me that Carlsen was Army Air Corps rather than a Navy officer, followed by a number which might be his service number. The second photo confirms this, with "Army Air Corps" spelled out in full on his own marker.
Q2. To resolve the confusion over whether Edinburgh Field aka Carlsen Field aka Carlsen Air Force Base is an Army Air Base or Naval Air Station, here's the history of the base on Wikipedia. The US acquired the rights to have a base there from Great Britain, and the base was used by the USAAF, the Navy and the RAF. Wikipedia says "in the fall of 1943 ... the 80th Seabees were brought in to build a station at Carlsen Field."
So -- to answer the question "who might hold records relating to operations of Edinburgh Field?" -- the answer might be all or any of these entities.
Some insight into the operations can be found on Scott Murdock's USAF INSTALLATIONS PAGE !. Scott published as part of his site a list by Joe McCusker of Air Force Bases. Note point 3:
A word about Army air fields, air fields and fields
Generally, Army Air Forces installations were named two ways: when
named to honor individuals they were designated "fields" and when
named geographically, they became "Army air fields." (The designation
Army air base was also used.)
Between the time the Air Force was established on September 18, 1947,
and the first designation of Air Force bases on January 13, 1948, some
Army air fields may have dropped the word "Army".
Carlsen appears further down on the page under a list of "Lend Lease" bases.
Of particular interest is the page Installation Research Tips, under the section Research Documents. Scott says:
General and Special Orders
Key events in an installation’s history are announced by the
publication of a special order or general order. The Department of the
Air Force (DAF) or the respective Major Command (MAJCOM) may publish
the order (sometimes both). These orders are important to an
installation's history much like promotion or PCS orders are important
to a military person's career—they are the "official" source of the
action they describe. Orders are not always perfect! They are
occasionally published with errors and may be revoked, rescinded, or
amended by later orders.
Two repositories that might be of help: the National Archives, and the Air Force Historical Research Agency. Their Welcome Page describes their holdings has links on how to make a FOIA request. For holdings at NARA, see the introductory page Records of U.S. Army Operational, Tactical, and Support Organizations (World War II and Thereafter) for their extensive overview.
As for Carlsen's military records, I believe what you can get via a FOIA request is limited unless you are a member of the immediate family. See NARA's introductory page to Research in Military Records. It may not be possible to learn more about the actual incident if the records are classified, but perhaps archivists at one of these two institutions could answer the question of what records are available to the public.
Other finding aids and oral histories, including this article Was It the Air Corps or Army Air Forces in WW II? can be found at the website of The Army Air Forces Historical Association.
Edited to add: Oh, wait! Those those personnel records are restricted for a limited time. Since Captain Carlsen died in 1942, his records should be available, if they survived. See this page at NARA:
Official Military Personnel Files (OMPF), Archival Holdings
- Records are accessioned into the National Archives, and become
archival, 62 years after the service member's separation from the
military. This is a rolling date; hence, the current year, 2013, minus
62 years is 1951. Records with a discharge date of 1951 or prior are
archival and are open to the public. Records with a discharge date of
1951 or after are non-archival and are maintained under the Federal
Records Center program. Click Official Military Personnel Files
(OMPF), Federal (non-archival) Holdings to access these records.
That page also has this link to more information about the 1973 fire at the NPRC and the efforts to reconstruct records lost.
See also the website http://www.armyairforces.com/ for their Research Help page, forum, and other resources.