You can get your OS onto your SDD for sure but you won't be able to get it off of your HDD without cleaning it, sadly. You can't have your cake and eat it too 🙁
If you don't mind reinstalling the games, however, you can just clean install both drive then reinstall onto your SSD. Then make your desired drive your primary for downloads and install your games back onto the HDD. That's the simplest and cleanest way to do it.
So, you CAN have your cake and eat it too, but it requires a fork, so to say. And that fork is your willingness to reinstall the games even though it may take some time...
The only other way I could think of this working out is expensive, however. You either get a huge flashdrive or a backup drive and throw the desired files you want to save on there or maybe even invest in another SSD/HDD for essentially the same thing, throwing all your files onto there. You can have your cake and eat it too, but you'll need to purchase the cake box first.
(Sorry I'll stop with the stupid metaphor....)