I am an electrical engineer and in the know about LED's,you could profit from reading my post "Lowdown on LED's" which contains useful info.I think what you are considering is certainly possible in principle and I may be able to help you in getting something built along these lines.If I myself had a setup like you, it would be pretty easy, but being not a fan of HT engines although I can appreciate your motivation,so I'm sort of up the creek without a canoe,but equipped with a paddle,or rather my electrical smarts.
The trick is to come up with a scheme that gets the 6 V battery charged without affecting engine performance unduly, secondly how to run your LED's from the battery without killing them( LED's are 2.5 to3.5 Volt devices and hooking them up directly to a 6 volt battery is a prescription for trouble, two in series across 6V will probably work OK ,so that's not the main problem.
It would be very helpful if I could get a better idea what that white wire source is like,electrically speaking.If I had that info I'm pretty sure could come up with a fairly simple circuit to stick between it and the battery that anybody with a soldering iron could build to accomplish your objective,but without a multimeter Volt/Amp/Ohms, preferably not a digital one -- the ignition circuit interference of the HT engine is likely to mess these up--- I would just have to take a stab at it & hope for the best.Let me know if you have any gear, some basic electrical smarts would be helpful too ,but not really essential.Let me know if you are game.