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I found the problem. The two files were forked from a single file a while ago, when it was decided that the client needed data separation.
Obviously, when files are duplicated the File ID (for File Authorisation purposes) is the same in both child files.
This seemed to confuse the authorisation process, prompting frequent requests for re-authorisation.
I went in to the security of each file, reset the Authorisation ID for the file, then set up File Authorisation again.
This had fixed the problem.