Unless I'm misunderstanding that is something you can do, I do it all the time as a matter of fact. You just can't do it all from the iPhone.
You need to AirPlay from your iPhone to your AppleTV. On your AppleTV select a different audio output. Thats it.
Sorry about the contrast but the above picture is my iPhone (blue box) AirPlaying a youtube video to that monitor on an AppleTV HD. The AppleTV HD has its audio set to a bluetooth speaker (red box). I use a set of Samsung headphones on occasion as well and it works fine.
You just need to pair the bluetooth speaker in settings > Remotes and Devices > Bluetooth.