CCC is a great tool and works well although the same applies, the Virtual Machine must be shutdown and preferably VMware Fusion closed while CCC is imaging it.
Since you have a newly created working Virtual Machine, sans user data, you can temporaraly attach the other Virtual Machine's hard disk drive to the new Virtual Machine and copy you're User Data over and then remove the temporaraly attached virtual hard disk. Have a look at: 
Add an Existing Virtual Hard Disk