Custom coordinate systems do not always publish coordinate system information to external files, such as a georeferenced PDF or Esri PRJ files when exporting to shapefiles. This is primarily due to the affine processor that is used in the coordinate system. In these cases you can still create the PDF or shapefile; however the custom coordinate system will have to be reprojected to a standard coordinate system prior to the export. You can accomplish this by simply changing the coordinate system and choosing to reproject the data.
In the case of the exporting to shapefile, the process and resulting files appear to be proper. It is when you attempt to import the results that you will see the problem. Here are some of the warning and alert dialog boxes and messages that you may encounter in these situations:
Bentley Map warning:
Import Complete (0 features in 1.103 seconds). One or more errors occurred, see Message Center for details. 0 feature(s) imported and 181 not imported for the [Feature Name] node. Failed to insert the provided instance with id = 0 and class name = [Feature Name] into the XFM storage. Unknown error converting source [Custom Coordinate System name].
Convert Local Coordinate Systems to Standard Coordinate Systems p20
Understanding Mapping Reference Frames for Engineering Projects p33