If it's a GOG version you've got, then simply copy-pasting the install works fine. Or you could of course download the installer and use that. If it's steam you can still technically copy-paste, though you'll probably have to run a file verification, which can take quite some time.
And yes, you have to download the whole game again, you should delete the early access client, it's not just a matter of patching it to save time.
