I had QT7 installed before installing FCS.
When FCS was installed it unlocked the Pro capabilities.
Your situation is that FCS is installed before QuickTime 7 is installed.
I would not have thought that the order of installation would have made a difference, but it seems to have done so.
The registration code for QT7 Pro is sent from Apple by email to those who buy the Pro facility.
As you are not buying it, you should not need a registration code.
Important:
Installing Final Cut Studio and entering the serial number automatically
unlocks the extended multimedia capabilities of QuickTime Pro.
The above is from the Installation instructions for FCS, and assumes QT7 is installed first.
Hopefully some one else can offer some help here, as I'm out of ideas, apart from re-installing FCS.
Anybody?