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In addition to manipulating the profiles, in order to use IMCE as a (kinda sorta) file field widget, the file field / custom entity must be able to delete files that the entity is managing. Again, I did this by adding an alter hook rather than changing any logic within IMCE.
This hook, hook_imce_delete_file_alter(), allows its implementations to remove files from the files-in-use list. Once that list is empty, IMCE allows the file to be deleted.