You do not need the Trial Serial number, if you have a full serial number.
But did you buy a full licence for Aperture 3 or an upgrade licence from Aperture 2? If you have a full Aperture 3 licence, the serial number you are seeing in Aperture from the main menu "Aperture > About Aperture" should be sufficient on your new Desktop.
But if you bought an upgrade serial number for Aperture 2 and not a full Aperture 3 licence, you also need to enter the Aperture 2 licence key.
If your current serial number does not work, contact Apple for a replacement serial number. You will need a proof of purchase, however.
In addition you can download the trail application from here Aperture 3.1 Trial
Installing from the dmg will give you better results then trying to copy the application from one system to another.
Once you have downloaded it use the serial number you paid for to authorize it, not the temporary trail serial number you received when you first loaded the trial.