you can get a very good 6v h4 led from the uk: the 5th gen one here:http://www.dynamoregulatorconversions.com/led-bulbs-especially-for-motorcycles-shop.php
you'll need the h4 reflector unless you wish to 'hack' it in the bpf reflector with cable ties (like my b33 - works fine but the light is less focussed). i went ocd on this topic somewhere with pics showing the various options -search the archives. my sr is still 6v and perfectly bright enough with the correct leds. and 6v charges at lower revs.
if you really want to go 12v replace the bulbs (front and rear), replace the regulator (fit a dvr2), replace the battery (a cyclon is my favourite) and that's it. no need to rewind the genny. the 6v horn will survive unless you drive like in india. don't buy one of those cheap leds with fans on the back. noisier than your bsa. i stress tested one for heat (like in a headlamp shell) and its lasted about one hour
Have you actually ridden with that double dipper at night.
My high pressure halogen parking bulb put out a better riding light than the very expensive, highly over rated but totally useless double dipper 6 V LED piece of lethal garbage.
It might work better on 12 V considering they are rated for 6 V to 48 V but mine was lethal at any speed over 10 mph on a road with no street lights.
By far the best 6V light I have ever used was a Hella 6V Halogen in an old Kombi waggon reflector which fits perfectly inside a Lucas shell.
The globe is the size of a golf ball with a very long element so you gest a really strong solid beam.
The Double dipper was good for reading a map 2' away from the headlamp but dangerous on the road.
You can tell they know that the globe is garbage by all the blame they direct towards the reflector thus making purchasers believe it is their fault that the light does not work.
Any one with 1/2 a brain should be able to see that a single CRE LED with a beam angle of 120 deg, sitting on a stick that throws a shadow over 1/2 the reflector can not work.
So all you get is the reflected light from that 120 deg cone bouncing off the reflector which accounts for a touch less than 1/4 of the reflecting surface of the reflector or 1/2 when both LEDs are energised.
Thus on high beam you get an elongated figure of 8 beam of light which throws a strong shadow of the front wheel over the ground it is about to roll over, no side illumination at the waist of the "8" and some feeble light in front of the shadow while all of the oncoming cars flash you because the light at the top of the "8" is in their eyes.