You can't use a capacitor to limit current once it has been rectified.
What you can do is put a capacitor in series with AC to limit the current, and rectify afterwards. It won't isolate from the mains, you need a capacitor rated to the full mains voltage (1.4142 times the rms) and the current depends on the supply voltage, frequency and the capacitor tolerance.
However, 12V and 300 mA is a lot for this sort of power supply. Also, the current though the capacitor will not be reduced with the addition of a small resistor, so you would be better to put all the LEDs in series so that the overall current is less. It is a good idea to put some resistance in series with the capacitor to reduce the in-rush current.
The impedance of the capacitor is 1/ωC where ω is 2πF so it is 314 for 50 Hz
For 300mA the capacitor needs to have an impedance of 240/.3 = 800Ω
So C needs to be about 4 µF. You can get capacitors of that size, rated to 250 V or more. They are called "motor start capacitors".
However, they are larger and more expensive than a small transformer that would supply 300 mA at 12 V.
I wouldn't use a series capacitor as a current limit unless the current is 50 mA or less.