Can I record iPad screen contents over AirPlay to an Apple TV and then recording the HDMI output?
There is a more elegant solution: Reflector

Unfortunately, it is a $15 app, but it serves a perfect function for your case. You can use your computer as an AirPlay receiver, and screen record the screen via ordinary computer screen capture software. No need for an Apple TV nor any cords, just a $15 app and a WiFi lan.