6.2 V zener shows lower voltage (Multisim)
I suspect you're not getting enough voltage across the bridge.  It's limited by a 100 ohm resistor and a 1uF cap, and then your diode drops kick in as well.  This is clearly intentional, since you don't want to zener regulate from 300V to 6.2V, but it's not particularly reliable since the current draw on the DC side will affect the voltage directly.  That (and personal safety) is why most such designs use cheap transformers or even complete wall warts.