You need to create additional Print Task service see the help below
We recently upgraded from 10.3 to 10.5. With the upgrade we lost our ability to print maps that contain secure services. Before the upgrade, we were relying on a workaround of creating a custom secure print service. According to support, it was determined this was a security risk and is no longer supported.
Support was unable to give me a way that it is possible in 10.5. This is a bummer since printing is a pretty important feature.
It looks like they will be reviving something similar in 10.6. Though seems like it would be much better to just allow the proxy to control this like it does for all other access.
You are right.
on the following link from help if you select 10.6 version providing the steps to create this service but not for version 10.5.
Also there is a caution about that (10.6 version)
It is not recommended to embed credentials in a custom print service without fully understanding the security impact. If you choose to publish your own service for printing with embedded credentials, it's recommended that you apply ArcGIS Server security rules to restrict who can access the service. This prevents anonymous users from generating printable map images that display your secured services. See Restricting access to GIS web services to learn more about setting up security.
A similar caution existed on version 10.3
Once you publish your own service for printing, it's recommended that you apply ArcGIS Server security rules to restrict who can access the service. This prevents anonymous users from generating printable map images that display your secured services. See Restricting access to GIS web services to learn more about setting up security.