Can you install the new drive into the PC at the same time ie before you remove the old drive? If so do the installation and then do a normal copy from the old to the new. Then remove the old drive or leave it installed for other use.
On the basis the drive to be replaced is a dedicated drive for X-Plane, copy the old drive X-Plane version to the desktop, which is the recommended location for running X-Plane, and then copy back to the newly installed drive. Again it is a normal copy function.
Alternatively copy X-Plane to a portable external drive and then copy back when the new drive is installed.
The bottom line is you don't need a new key and no need to run installer. You can have as many copies of X-Plane as you wish on your PC provided you run one copy only of X-Plane at one time and use the same key originally provided at the time of purchase.
Hope this helps