CCC is NOT an alternative to Time Machine and shouldn't be considered as such. CCC is an invaluable piece of software and I wouldn't want to be without it, but it cannot work in the seemless way that TM should. Both products are brilliant at what they do, but they are not interchangeable.
So you need to sort out your TM setup. Since you are not using a true TM supported backup device, that's the first thing to check. What you are using may well have a problem with Moutain Lion that did not exist previously. Not necessarily a fault of either, but these things happen as software that is supposed to work together, but is supplied and upgraded from totally unrelated developers.
As suggested, connect a backup device directly. A USB HD would probably be the simplest and see if that works as expected. If it does, then it points unequivocally to your WD MyBook Live as the problem. If it is still so slow, then you need to look further at your OSX installation. But I suspect that's not going to be the case and it's an incompatibility between your WD MyBook Live and ML.
Before anyone sounds off against Apple for bad development of their OS, remember it is not their responsibility to check it againt absolutely everything else out there in the marketplace as that would clearly be an impossibility. It is in truth up to the third party suppliers to ensure their product is kept up to date with developments of the OS with which they claim their product is compatible.