Ok, time to go through some tips about this issue.
I tried a different battery in my 4S and it completely got worse, so I switched back to my old battery. If Apple replaces your battery, torture your phone as much as you can before the waranty ends and get a replacement.
So for now, I'm somewhat "satisfied" again because I did this:
1. I j....roke my phone and have a killswitch for all running apps now. This makes quitting apps a quick swipe. Or do this manually by doubletapping the homebutton and swipe the app-image upwards out of the screen. Repeat this step as long as there are apps open. Might take some time, depending on how many apps you are using.
If anyone from Apple reads this: I'm still waiting - the length of this discussion is clearly showing that you have homework.
2. Go to your "privacy" settings and disable all location services you don't need. Does FB really need to know where you are? Please note: that there are some "hidden" Services on the bottom called Systemservices. I disabled them ALL Please be advised: There might be some features, that are not working like you are used to. I didn't notice any difference.
3. I reduced my screen brightness a bit.
This way I got back to seeing the 10% warning again.
It really looks like the sensor for emergency shutdown in iOS 7 is quite sensitive and can't cope with larger energy demands of flawed batteries. So replacing the thing MIGHT also help you out, but as I said in the beginning: TEST this thourougly. In my opinion iOS 7 ist still beta...