CCC can be used to help with the transition to a new hard drive. We have found that formatting the new drive after installation can be more reliable than formatting in an external enclosure or using a drive adapter, these devices can cause the drive to only boot when in the enclosure or with the adapter. 

After installing the new drive, boot from your CCC backup or the old internal in an enclosure or using an adapter, then format the drive as we describe here: Preparing your backup disk in macOS.

Once formatted use CCC to restore the data from the booted source to the new drive as covered in this article: How to restore from your backup.