is the name of an airport such as LAX the act of the FAA or the ICAO? from reading, it looks as though the process starts at the ICAO as a recommendation, then is adopted by the FAA?
4-letter names are too trivial to be protected by copyright. Trademark protections require 1) specific applications, 2) commercial usage, 3) that the offending use is relatable enough to the original business to dilute the trademark or harm the business.
(Thus, for instance, an "Airbus" car would be a clear a trademark violation, for a model plane called "Airbus A320" some agreement is preferred just to be safe, and for a book titled "Airbus A320" no communication with Airbus is needed, as it does not compete with any Airbus product, nor does such a title on a book imply being one.)
IATA and ICAO codes amount to nothing more than telephone numbers. They cannot be copyrighted, since they represent abstract concepts rather than a creative expression of one, and they aren't trademarked.
More than that is a question for Law.SE.