I was wondering then, as I don't have a Time Capsule, whether it would be possible to dedicate the second HDD for exclusive use of Time Machine, backing everything up?
Anyway, would splitting the two work and would it be easily achievable with existing tools in OS X?
Yes, you can use the 2nd HDD with Time Machine as your TM back up drive.
As long as you are able to format your new SSD and the HDD, you can use them independently. I would advise using Terminal and the diskutil commands to do this as Mountain Lion 10.8.3's GUI Disk Utility will see the SSD and HDD as needing to be "fixed" (drives marked in RED) and if you choose to FIX the drives your two drives will automatically be setup as a Fusion drive and all data on the drives will be lost.