Restore from 2014 Time Machine backup to Old 2013 hard drive

I have a 2007 MacBook Pro. In January of 2013, I used Disk Utility to clone the 320GB hard drive onto a new 500GB hard drive, removed the old 320GB hard drive, and installed the new 500GB hard drive. The new hard drive worked perfectly for 14 months. I kept the old 320GB hard drive in a box in my office, just in case I ever needed it.

(Also, you should know that when Apple released Snow Leopard years ago, I upgraded all of our family computers by downloading 10.6 from the Apple website. So I do not have a Snow Leopard installation DVD. The last installation DVD I have is Leopard, 10.5.x)

Now the 500GB hard drive has damaged sectors involving Snow Leopard 10.6.8 files. So I can't boot from the drive. I have a February 2014 Time Machine backup of this drive. My Time Machine backup is on a 2TB external hard drive that is exclusively used for Time Machine backups of all our family computers.

I have put the damaged 500GB hard drive in an external enclosure, and attached that to my iMac which also runs 10.6.8. I used Ddrive Genius 3 to find out that the drive has damaged sectors.

The iMac's Disk Utility recognizes the damaged external hard drive, but the Restore function is not available, it's grayed out. Drive Genius will not restore from the damaged hard drive. Also, when I go to System Preferences, Start Up Disk and try to choose the damaged drive as the boot drive, System Preferences does not find it. So on the iMac, Disk Utility and Drive Genius see the damaged hard drive in an external enclosure, but Start Up Disk does not see it.

I have also put the old 320GB, working hard drive into a 2nd external enclosure, and it is also attached to my iMac. Disk Utility recognizes it, and I can boot my iMac from it using Start Up Disk. Since Disk Utility sees it, I could use Disk Utility to Restore from it if I needed to. But if I did try to Restore the damaged 500GB hard drive from the January 2013 320GB hard drive, I would lose 14 months worth of data and apps. This is not an option.

When I boot the iMac from the 320GB hard drive dated January 2013, and attach the 2TB Time Machine backup hard drive to the iMac, the iMac recognizes it, and Time Machine will open and show the backups that are on the 2TB hard drive, including the backup of the damaged 500GB hard drive from February 2014.

But Time Machine will NOT restore the February 2014 .sparseimage backup to the January 2013, 320GB hard drive, although there is room on the hard drive to do this. The "Restore" button is grayed out and I cannot get it to restore. I'm guessing that since I am booted from the January 2013 drive, Time Machine is confused and wants to restore from a backup dated before January 2013.

I have been afraid to erase and format the damaged 500GB, because if I am not able to restore from the Time Machine backup, I will lose 14 months' work. Again, I do NOT have a Snow Leopard DVD to re-install from.

I have been spending every spare minute for 2 weeks trying to figure out how to save/restore the data and apps from my damaged 500GB hard drive.

Any help will be deeply appreciated.