Unfortunately, to add support for MIDI you have to hack in the Arduino core files. I've hacked in a MIDI-USB implementation in an alternative set of core files, which is available at GitHub - rkistner/arcore: MIDI-USB Support for Arduino. The MIDI-USB protocol is fairly simple (once you get around the USB interface descriptors), and the implementation is quite similar to CDC (I based my implementation off that).
In the long run it might be better if we could use an existing stack with MIDI support, but it seemed complicated to replace Arduino's USB stack with another one.