It's difficult to see any other course of action you can pursue other than fresh-installing the OS onto your 128 GB SSD and then reinstalling your programs onto your 1 TB HDD. I realize that probably doesn't hold much appeal for you but I don't see any other viable option.
While cloning the contents of your 600 GB HDD to your 1 TB HDD poses no problem, that process will only be effective insofar as your personal data & other non-program files are transferred to that drive. But having a clone of your current HDD is a good thing since it will act as a very desirable comprehensive backup of your system. And that's important in the event things go awry in the future with your SSD. So by all means do pursue that cloning operation.
But understand that you will need to install your programs on the cloned 1 TB HDD. The cloning operation does not posses the capability of actually installing the programs on its drive. (I'm assuming, of course, the disk capacity of the 128 GB will not be sufficient for you to install the programs on that boot drive. But perhaps you'll have sufficient disk space to install some of them.)
There should be no problem with the fact that both the SSD & the 1 TB HDD will each contain an OS. Just ensure that when you install the OS onto the SSD the 1 TB has been disconnected from the system. Also, connect the SSD to the 1st motherboard's SATA port - probably designated SATA 0 or SATA 1. Of course ensure that the BIOS boot priority order reflects a first boot to the SSD.
Work with the SSD for a reasonable amount of time to ensure it's functioning properly. Then connect the 1 TB HDD. Do NOT make any significant changes in the data on the HDD until you're reasonably certain the system is properly functioning.