^^^^^^^^^^^^^^^^
this.
this could be your best option. you'll have to check the circuit and see how it's switched. if it's just a relay, throw a bridge on there and a 2000 uf cap on there and you'll never know it's been modded. I wouldn't bother with a regulator. for a 12 volt regulator you need about 16 volts (at full load) to get proper 12 volt regulation.
12 volt car LED's will be good from 9-15 volts (as charging systems can vary) i'd just rectify 12vac to 12vdc and smooth with a cap and whatever you get, call it good.