Q: Is there a way I can watch video on my Apple TV via AirPlay but listen through headphones directly connected to my iPad? Whenever I sent a video to my Apple TV via AirPlay, the audio goes there too. Can I just sent the video only? Can I send the audio to a set of Bluetooth headphones or Bluetooth speakers instead?
A: The short answer is no, at least for a stock iOS configuration.
As you’ve noted, AirPlay transmits both audio and video across the same channel, and unlike AirPlay audio in iTunes on your Mac or PC, there is no way to send audio out more than one channel at a time.
Note that you can stream your audio from the Apple TV via AirPlay to another device, which would allow you to use a set of AirPlay speakers as an alternate audio output, and this works even when streaming video from an iPhone or iPad.
Further, there are no headphone options specifically available for AirPlay; an AirPlay option for headphones would likely be considerably more expensive and less versatile than traditional wireless headphones that you could simply attach to your stereo system via a base station.