the voltage drop across the diodes in the bridge will reduce the voltage as the voltage drop is burned off as heat, i would for sure heat sink this application.

If you see the big power diodes on the squawk and talk board and even the cheap squeak. They run the 12v through a few power diodes just to get that voltage drop across each diode. They wanted to burn the heat off some extra heat at the diode instead of the regulator. Too bad on the squawk and talk sound board, those diodes just dry up the 10uF cap above them and cause more problems than the diodes where trying to fix.