Once you have copied the ExportWebmap GP Tool as GP Service and configure the credentials on GP Tool, you can use the Url to this REST endpoint instead in your PrintTask.
In the steps you referred to, only "user connection to ArcGIS Server" is mentioned. The WMS services we're using is hosted by another party, so we can only access it from ArcGIS for Desktop by adding it as a WMS Server (http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//006m00000062000000.htm), not an ArcGIS Server. Will it work anyway?