[Scuderia_Turini]

Hello Kev,
Well the 3.3volt regulator idea is basically not on.
The advantage that white LEDs operate at 3volts is actually the downfall.
The operating voltage is too close to the stored voltage of the super-capacitor, so pretty soon the white LEDs go out, although the red ones stay lit for about 2.5 minutes.

The video shows a 5volt source feeding the 3.3v regulator (in the clothes-peg) a diode then feeds the 3.3v to the super-capacitor and LEDs.  
The white LEDs are driven directly and the red LEDs have 1.5K ohm resistors in series (to dim them to a realistic level).
The diode is necessary so that the super-cap is not feeding back to the power source when the power is cut.  The diode drops 0.6v so the capacitor gets just under 2.7v and that is marginal for keeping the white LEDs lit so once the capacitor has discharged a bit the white LEDs go out.
 
For the video I wired in a green LED with the white ones and you can see that it stays lit for a bit longer. So if green headlights are your thing...

A 5volt regulator would work better but those tend to be much larger.

After I mentioned my F1 Medical Car, I remembered that for those I install a 12volt battery to power the lights. An N size battery will fit in most saloon type cars and operate for quite a long time. Of course that carries a weight penalty. An N size battery is same diameter as an AAA and 2/3rds it length.

Leo