Currently, there is no equivalent process in ArcGIS Pro to embed the credentials in a custom print service. Please add this functionality to ArcGIS Pro.
We are creating print services from ArcGIS Pro, following the documentation here: http://enterprise.arcgis.com/en/server/latest/create-web-apps/windows/tutorial-publish-additional-layouts-for-printing-with-arcgis-pro.htm
We need to embed the credentials in the print service so it can be used to print secure services. The only way to do this currently is in ArcMap. The documentation for printing secure services in ArcMap can be found here: http://enterprise.arcgis.com/en/server/latest/create-web-apps/windows/printing-maps-that-contain-secured-services.htm. There is currently no way to apply this to ArcGIS Pro. This is forcing users to continue using ArcMap, rather than being able to fully migrate to Pro.