Restore from "iOS7 backup" to an iOS6 phone
You can edit the info.plist file. I successfully restored my backup to 6.1.4. Edit the build version to your ipsw's build version and product version to the version you want to downgrade to. Using plist editors in windows are highly recommended.

Windows: C:\Users[Your username]\AppData\Roaming\Apple Computer\MobileSync\Backup[Your UDID]

Mac: ~/Library/Application Support/MobileSync/Backup/[Your UDID]/

After that just restore the backup normally. If an error popped up at the end, ignore it. Hope this helps.