I have never been able to figure out how to access CDDB directly from Cover Designer, presumably because it can't be done(?). So, like you, I do the work-around of accessing it through Nero and then launching Cover Designer from within Nero.
In order to get your trick to work with for a simple disc copy, open the "File"->"Preference" menu and select the "Database" tab, then make sure the "Open the database during 'CD copy'" box is checked at the bottom of the dialog. While you are there you might want to make sure all three "Open the database..." boxes are checked.
When select you "Copy" from the disc copy dialog, Nero will begin the copy process by analyzing the disc (getting the disc information by reading the first few sectors), and then Nero will open up the database interface and allow you to select the CD and print the CD cover.
After you have opened up Cover Designer by pressing "Print CD Cover" in the Nero dialog, you can either continue with the actual disc copy in Nero or cancel it if all you really wanted to do was print the cover.