Your laptop will fill up quickly, so having another drive makes a lot of sense.
Unfortunately, the answer to your question is typically no. You can’t just copy the folders from the old location to the new. As you suggested, you need to perform a complete uninstall and then a reinstall, specifying the new installation location when you set up the program again.
Let’s look at why that is and an exception to it.
When the game was installed, its setup program wrote information about the game to Windows, the registry, and a few other places. This information is typically used by Windows and by the game to locate parts of the game, use shared components within Windows, and may even include things like your game’s registration information.
Just moving the games files doesn’t change or update any of that information; all it really does is invalidate a lot of it.
Trying to run the program from a new location without that information being correct means the program won’t run – in fact it’ll probably just crash when you try.
The rule of thumb is actually pretty simple: if you had to run a setup program to install that software, then you need to run that same setup program to either move it or uninstall it and then reinstall it to a new location.
The other side of that rule of thumb is pretty interesting: programs that don’t require a setup.
These applications, called portable programs, don’t use an installation program, but instead just “deal with” the state of the machine when they’re run. If something’s missing, they’ll just create it or update it as needed.
In reality most portable programs try to have as few dependencies on information stored on the computer as possible.
As a result, much of what I said above about programs that use a setup doesn’t apply. They will actually react very favorably if you simply move the folder or files.