In follow-up to this, I did some testing together with Milos. The following results:
FBX with camera export enabled: works OK
FBX with camera export enabled as part of mesh: “file not found” after saving FBX and model textures. No camera texture and no info file.
OBJ with camera export disabled: works OK
OBJ with camera export enabled and PNG texture: “file not found” after saving OBJ and model textures. No camera texture and no info file.
OBJ with camera export enabled and JPG texture: “file not found” after saving OBJ and model textures. No camera texture and no info file.
Conclusion: the export fails when trying to export the camera texture.
Console screenshot of all four export attempts: