Direct Music Player audio to Caller
You cannot

Media Playback developer documentation says (emphasis added)


  You can play back the audio data only to the standard output device. Currently, that is the mobile device speaker or a Bluetooth headset. You cannot play sound files in the conversation audio during a call


Edit

In response to your edited question:


I am not aware of Cyanogenmod permitting this- searching in their source code may give a clue
To enable this, I presumed would require programming, so searched Stack Overflow and found this Why it is not possible to play an audio file on a voice call in android. This gives a couple of reasons, in answers as to why it is not possible - security and prioritization of audio
It also suggests ways of  achieving which are in programmers domain