An answer on your idiot light question is no, you can not replace the lamp with an LED. You can replace it with a resistor but that doesn't help what you want to do. The idiot lamp serves another purpose. One side of it is tied to the IGN line off the ignition switch. The other side of the lamp is tied to a field output of the alternator. When you first turn on the key, the field gets voltage from the IGN line through the lamp filament. This voltage energizes the field so that the alternator will start producing an output. Basically a LED won't work.
There are a few ideas floating around out there to do something else besides an idiot light. I haven't given it much thought. Off the top of my head there are circuits out there that switch off items when voltage falls to a certain point. Some of these circuits are used in RVs to cut off power to things like refrigerators if the battery gets to low. May be able to see something like that on a RV site or maybe out of JC Whitney. Modify the circuit to switch at around 12.5 volts and activate your buzzer.
There are a few ideas floating around out there to do something else besides an idiot light. I haven't given it much thought. Off the top of my head there are circuits out there that switch off items when voltage falls to a certain point. Some of these circuits are used in RVs to cut off power to things like refrigerators if the battery gets to low. May be able to see something like that on a RV site or maybe out of JC Whitney. Modify the circuit to switch at around 12.5 volts and activate your buzzer.