Goldfingas
Is the music that you have locally downloaded from Apple Music? And there are ways around that on the Mac. You can use programs like loop back, or audio hijack to get the job done. I don't know that you'll be able to record your sets with the phone unless you connect the phone to a controller that hasn't built-in audio interface, and you record from the outputs of the audio interface into either another computer, or figure out a way to capture the audio coming out of your audio interface.