You can do this with the API, but I don't think there is a built-in option that will allow you to create populate the variables in a template by right-clicking on a document to create it.
In therory you can achieve this using Dispatch... it will not be a complete solution.
You can create a Dispatch script that you set to run via the menu. Select the "Drawing" and then run the dispatch script. In the Dispatch script you can copy a template ECN file to a new location. You can also read datacard variables from the "Drawing" datacard and set them to the newly created ECN file datacard (CREATING THIS PART OF THE SCRIPT IS A PAIN...). The ECN file can be checked in where EPDM could populate things like "ECN Number" from a serial number on the datacard.
Naming of the ECN in the Dispatch script can be done using information from the "Drawing" but not from the serial number generated from EPDM unless you run a second Dispatch that runs 'after checkin' or during the first 'automatic transistion' in the ECN Workflow to rename it.
The ECN will not be linked to the Drawing file... if you want a complete solution then you'll have to have someone write an Add-In.