Yes, your OS should work OK on the SSD if you make a direct clone of it from the hard drive drive.
My normal backup routine is to clone my main system drive, which is an SSD, to a hard drive and I can boot from either drive.
If my SSD dies or becomes infected/corrupted I reboot into the backup hard drive and make a clone copy of that to the SSD.
Never had a problem with Windows complaining about an invalid key.
Note that I CLONE the drives, not IMAGE them.
Cloned drives are bootable so work well as backup drives.