The modern method for reinstalling Windows 7 is to reimage the HD or its replacement using the built-in backup imaging, from an image you capture just after install, or in your case additionally after Anytime Upgrade. Backup Complete Computer - Create an Image Backup System Image Recovery
I agree it's unlikely you can recover the Home Premium keys now and should focus on recovering them from their source as they are possibly recorded somewhere and traceable by confirmation emails, receipt, bank info, etc.
However since Anytime Upgrade only unlocks extra features while overwriting the Product ID and Key with the new ones, it is worth auditing the info with Belarc Advisor
to see what it finds.