It seems to me that you forgot to take into account the effects of the smoothing capacitor or alternatively you have very low current output from the power supply.
Basically, the peak voltage of that 13.5 power supply is going to be 13.5*sqrt(2), which gives us around 19 volts. Two diodes give us around 1.5 V, so the peak is 17.5 V. This gives us plenty of room for a 12 V regulator, unless you run the supply at its full output current. So what you can try out it to make a rectifier, place a big enough smoothing capacitor after it, make a dummy load from a 150 ohm resistor (make sure it's 1 W or higher!) and see the voltage you have on it once it's connected. If there's room for a regulator drop-out voltage, then use a regulator too.