We have a 12 hour layover at the Beijing International Airport

You’d better save two hours at least for boarding your flight and then you may have ten at most to visit the Great Wall and the precondition is that you go there on a car. If you want to take the public transportation to the Great Wall, the time may be very tight and you are advised not to do that. And taking taxi is very expensive. Meanwhile, I will recommend you to visit Mutianyu Great Wall rather than Badaling Great Wall because it generally is crowded in Badaling and may takes too much time on the way.