It might be related to 1.) how you load the files and 2.) which changes you perform.
If you load the files individually, e.g., by using drag and drop from Finder or Music.app, Mp3tag gets secure access to the files but not to the actual folder of these files. If you then perform a change that requires secure access to the file's folder, i.e., renaming files or performing changes that require a temporary file (e.g., adding large cover art), Mp3tag runs into that issue.
So if you load the files via the folder or maybe even a higher level folder (e.g., ~/Music/), Mp3tag gets secure access to all its containing files and remembers this via a secure bookmark.