Anytune does not currently have a way of recording a LiveMix session from within Anytune itself.
We had intended to do this, but we’ve been focused on other features. What you can do is make use of Audiobus to stream audio to an Audiobus enabled DAW or recording app like Audioshare. That will let you record the combined Anytune Music + LiveMix output.
We have an overview video of how to use Anytune with Audiobus and an (admittedly) older version of GarageBand here: