Some things that might help:
replace the 1n4001 with a 1A schottky diode. The Schottky diode switches much faster than a 1n4001 which might help if it is the inverse spike that’s causing the second trigger.
Put a 50pf cap across R1 which will speed up charging the gate capacitance of the mosfet and make it switch faster (you may need to experiment with higher or lower values to see what works best)
Swap the 2n7000 for a higher current / lower RDSon value mosfet The 2n7000 is 5 to 6 ohms max RDSon (you still need the 3V gate threshold though). You look to maybe be drawing more than the max 500ma of drain current depending on what the LED voltage drops are. 15 ohms at 12V will be about 750 ma or so
With added circuit complexity you could also add a one shot at the junction of the diodes and R1 so you can stretch the length of the trigger pulse (a 555 timer should do) which may increase the amount of light you get if the current trigger is too short or it may not help at all but may be worth a try. Basically the coil/diodes trigger the 555 and its output drives the MOSFET for a fixed time. For testing a pot then allows you to stretch the pulse for more light output to see if it helps (or burns out the LEDs of course if it gets too long 
).
Peter