Simming it with a silicon bridge rectifier, it doesn't look too bad with enough good low ESR capacitance but that bridge will kill your headroom. Due to the supply current being concentrated near the peaks of the voltage waveform, I'm seeing about 45A peak through the bridge for 12A average DC current. That's only 20.5A RMS so well within your bridge's ratings. However, if you look at Fig.3 of its datasheet, on the 35A curve, the instantaneous forward voltage drop
PER DIODE is 1.2V. That means you are loosing 2.4V that you can ill afford in the bridge. For comparison, the sim says the Motorola MUR2500 50 Volt 25.00 Amp bridge I have a model for drops 0.9V per diode. A good set of Schottkys can save you about a volt of total drop in the bridge, headroom you badly need if you want to push above 12A, or when the mains voltage drops when half the country puts the kettle on in the commercial breaks in Corrie or at the end of East Enders.
Its looking like you need 20mF minimum capacitance to keep the ripple trough above 12.5V so you've got enough headroom to regulate. However I'm seeing 16.7A RMS ripple current in that cap for a 12A load so you are going to need a bank of them to split the ripple current up to something more reasonable so they don't puke their guts at full 12A load current. The best deal (price per mF) on decent quality caps I could find at Mouser was
http://www.mouser.co.uk/ProductDetail/United-Chemi-Con/EKYB250ELL682ML40S, (25V 6800uF, 4.22A ripple current), and you'll need at least six in parallel to meet the ripple current spec. Eight would give you some margin, but you might as well go the whole hog and use a bank of 10 as there's a price break at qty 10.
Recalculating with 68mF capacitance, using 1/10 the worst case ESR from the above cap, and assuming 0.05R wiring resistance between the bridge and the caps, it looks like you can maintain 1V headroom at 15A load current. The ripple current is 24.6A RMS, well within their rating.
Unless you've got some server grade 25V caps hanging around, it looks like it will cost you £13.50 for the caps. They are going to need decent busbars as well - probably the easiest option without getting a pcb made would be to mill a break in the copper of a piece of single sided copper clad, drill it for all the caps and sweat solder two heavy gauge solid coper wires to it to improve its current carrying capability.