I'm having (I think) the same problem, though I'm running an older rev of Cover Designer (188.8.131.52), which came with Nero 184.108.40.206. I've tried using all the paper stock setting options you've mentioned: File>Paper Stocks..., File>Preferences>Paper Stocks, and the drop-down menu you mention in the toolbar (which is indeed present even in my older rev).
The exact behavior is that the pop-up works, but the list of stocks in the pop-up is only a subset of the list shown in File>Paper Stocks... (problem 1), and of course, the stock I want to use (Avery 8931 CD Labels (Letter)) is not in that subset; when I select that paper stock in File>Paper Stocks..., according to the manual, that should select that stock for use with the current cover design, but the selection in the pop-up list does not change to match, and when I go to "Print...", it selects what's in the pop-up (problem 2).
In the "Print..." dialog, the stock selected in the pop-up does appear in the "Elements" tag, but if the first element ("Booklet (Front)") is un-checked, it reverts to "Normal Paper" (problem 3).
I tried defining a "User-defined Stock" in File>Paper Stocks..., and I find that it is also not available in the pop-up list in the toolbar (problem 4).
Is there some way to force a re-load of the list of stocks in the toolbar pop-up list to include everything in the File>Paper Stocks... list? Or is there some other workaround to get either a User-defined or the Avery 8931 CD Labels selected for printing a CD label?