This may be the same issue we are having, if so it is a bug, but not on ESRI's priority list http://support.esri.com/bugs/nimbus/QlVHLTAwMDA5NjkxNA== Update from the analyst is below. - BUG-000096914 - Printing an external secure service with a limit usage on referrer URLs fails in the ArcGIS Online map viewer or application As you mentioned, closed as a duplicate of BUG-000098312 - BUG-000098312 - Printing service fails to print an ArcGIS Online or Portal for ArcGIS item that secured with limited access and credentials are embedded within it. Closed and fixed in 10.5. There was no additional information in bug notes. Therefore, you are correct, BUG-000096914(BUG-000098312) has a fix in 10.5. In the meantime, ArcGIS Server Manager Upload Patch does mention the fix for BUG-000098312. However, since there is no version 10.4, the customer will need to upgrade ArcGIS Server to 10.4.1 to apply the patch.
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