I'd simplify and remove the regulator entirely. Maxwell makes a 16V 58F cap module. One of those will supply two amps for fifteen seconds with roughly half a volt drop. You don't specify a tolerance on the 12V so I'm guessing that's acceptable. You still need a current limited source to charge the cap, but a 100W wirewound resistor would be a good guess. Put a small schottky around the resistor to discharge back to the load. Easy.
