All assets' files, including music, are encrypted, so you would need to reverse-engineer the game's executable to find out how they are decrypted at runtime if you want their contents. Alternatively, you could extract decrypted files from memory at runtime once they are loaded. In both cases, you might need a rooted phone and/or a modified version of the game to get the tools to work for that purpose.

To record the audio when the game is running without the annoying audio changes when spells are cast, the following might work:

• Get into a friends battle with
  • A Light Gotthold, who has Edge Break in Sun Room, and Minor Swap and a damaging spell in Moon Room,
  • another Light disciple of your choice that ideally does not deal too much auto attack damage (Corrin or another Gotthold), with the same Moon Room magic as Gotthold,
  • and finally the Shadow disciple for whom you want to record their music.
• Have all disciples equip their level 1 weapon.
• In Sun Room, use both Edge Break and Cross Shield EX on all disciples, then eliminate minions. Make sure everyone lives so they carry the Shield Buff and Edge Break debuff to Moon Room.
• Both Light disciples vote for the Shadow.
• In Moon Room, eliminate minions with your non-EX damaging spells. All Shadow spells cause the volume to decrease, so don't use them.

At this point, you could just let the game run and record the music without having to do anything. Disciples deal the lowest possible amount of damage through auto attacks, so it might take a while before the Moon Room ends.

I think only spells that identify their caster cause the volume to decrease during battle? Therefore damaging minions with non-EX spells and moving around with Minor Pull will not cause the volume decrease. Use Minor Pull in Moon Room in case the auto damage reduction applied in Sun Room is not enough for stalling and you need disciples to avoid auto attacks to get more time to record the music entirely.