I'm not sure that's possible with Breakaway Audio Enhancer. If I'm not mistaken, BAE won't send its processed audio to another application by design. This is probably to keep people from using it as a broadcast audio solution and encourage them buy BreakawayOne. I don't have an Apple TV or airplay compatible device, so I can't test it out directly, but I'm willing to bet BA1 can do what you want with a little configuration.
I suggest trying out BreakawayOne in trial mode to see if it can do what you want. You'll be impressed with the improvement in audio quality and the jingle/nag doesn't interject often enough for it to be a real issue for personal listening. I ran it in trial mode for a number of months before I invested in a BA1 license.