Although my first port of call would be heat shield, I would also ask: How old is your battery? I am a bit suspicious of your <12 volts measurement, as a reasonable battery should give >12volts without help from the alternator . It is possible that there is an internal short in the battery, maybe due to buckled or sulphated plates which places an additional load on the alternator. If you still have the same battery since you bought the car, there is nothing lost in renewing it anyway as it would now be past its useful life.
