Need help choosing options for 12V supply from a Lithium-Ion battery
The problem is, all the modules I've looked at have a maximum supply
  limit of about 14.5V and worse, their sound quality starts to diminish
  at about 10.5V.


A good 3S Lipo has a usable range of 10.5-12.6V when discharged at a reasonable rate (1C or lower), so it should suit your application perfectly. 9.6V is into the over-discharge zone, where the battery has no capacity left. You should never let the voltage drop below 10.5V under load (11V resting).


  So with this choice, the sound will begin to suffer long before the
  pack has exhausted its useful charge.


If the amp sounds good above 10.5V then by the time you notice the sound quality degrading the battery is already over-discharged and will die very soon. You should use an under-voltage alarm and/or cutoff device to prevent the battery voltage from going below 10.5V.