The problem is you haven't purchased an activation code. You've purchased a full version. We use third party vendors for both these elements, so it's going to be much more difficult and time consuming to try to get your money to the right place to process a code for you.
You can verify this easily. First, you're looking in the wrong place for your full version download. Instead of Download Center, open My Order History. When you look under Activations, you'll see it's empty because you never purchased an activation code.
Under Store (full version) purchases you'll see PI 6.5. Click the Details button. There, you'll see your link to a full version download which you haven't ever actually completed. Click that to download your full version.
Be sure you've uninstalled the PI6.5 version you have currently installed first, as noted before. I also recommend restarting the machine, and verifying that the Program Files/ArcSoft/PhotoImpression 6 folder is really gone after doing that--if it isn't, manually delete it.
Then install the full version from the noted download link.
I haven't the faintest idea how your backup disc could possibly be a trial version. You'd have to contact support again to discuss that. Also, if you have any trouble with what I've mentioned above, support is the place for further assistance. I recommend phone support since those reps are in this office and will be able to assist more quickly with such problems. (I'm sorry they apparently steered you wrong before, but can't do much about that now--they remain the best option.)