If possible, I think you could go to visit downtown Beijing. After arriving at Beijing Airport, you could take the Airport Express Train to Dongzhimen and then transfer to Line 2 to Qianmen station. It will take you about 40 minutes. Then, you could have a breakfast at Qianmen Street, where many time-honored restaurants are located. Then, you could walk to visi the Tiananmen Square and then a 2-hours brief trip to the Forbidden City. You could finish the short trip around 10:30 and then take the same way or a taxi back to the airport around 11:30~12:00. At the airport, there are many restaurants for your choice to have a dinner.

Ps: if you worry about missing the flight due to the large Forbidden City, please pass it and visit these attractions around the Tiananmen Square.