Unfortunately there really isn't an "elegant" fix for this. You could try dimming your screen a bit more before going to bed, or try using a darker wallpaper. I always use a black or very dark colored wallpaper anyways since it's easier on the battery. So when it does come on in the middle of they night, it won't be as blinding. I'm usually asleep by the time my S5 finishes charging, so the 5-10 seconds the screen does lights up, doesn't usually bother me.
That said, I do usually wake up each morning by the light of my screen, before I actually hear the alarm. But I have my alarm set for 2 minutes before it reaches full volume.
This way my alarm doesn't wake up my wife, if by chance she happens to get up after me.
Edit: I just realized.....have you just recently upgraded to Lollipop? Obviously, that would explain the bright white notification bar, as opposed to the black bar on all KitKat devices. My suggestion above may be a moot point if this is the case. One of the many reasons I never took the update.....I can't stand the over abundance of white that was incorporated into this update.
If my suggestion above doesn't work for you, you might look into a third party launcher. There are some really great ones out there (I use Apex Pro, but there are others that are equally as good, if not better) and this way you can get rid of all those white backgrounds, including the notification bar.
Good luck and let us know how you make out.
S5 tap'n