Hi,
as FET try a FDV304P.
It seems you made some changes on the circuit... parts, voltages, values....
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If you want a LED that tells you when it is out of regulation (less than the desired 1mA), then use comparator and a 3.9V zener: cathode to VCC and a 10 kresistor form anode to GND.
Use a comparator and compare zener voltage (connection between 10 k and zener) against OPAMP output voltage. As long as OPAMP output voltage is higher everything is OK.
When OPAMP output voltage is lower then it is out of regulation...
Klaus