I think FATE music should only start playing if you are participating in the FATE (attacking FATE target), the same way The Hunt music works.

At the moment you can be riding around on your mount, enjoying the mount music, but if you ride through FATE area boundaries the music gets interrupted and changes to FATE music. It can be hard to avoid passing through active FATEs when travelling.

I think the same should apply to the large text FATE Commenced / Complete / Failed. It doesn't need to splash these messages on the screen unless you are participating in the FATE.

FATE music takes such precedence at the moment that you can be fighting your weekly hunt mark which happens to be near a FATE, and it will play the FATE music instead of the B rank mark music while you are fighting it, even though you aren't doing the FATE.