Actually the whole 12v charging system is very easy.
You need charging coil that is NOT grounded at one end. It looks like the one offered by Gasbike is the easiest to modify:
https://www.gasbike.net/collections/magneto-electric-parts/products/12v-generator
See where it's soldered on the top end? Unsolder it and attach an insulated wire and run it outside the case with the other wire. Then attach it to a cheap 4-pin moped/motorcycle rectifier/regulator like this $4 one for a GY6 moped:
https://www.banggood.com/4-Pin-12V-...-125cc-150cc-Moped-Scooter-ATV-p-1134889.html
The two wires off the coil are the yellow wires, and the red goes to the battery + and green to battery - (ground).
For a battery you can use a small lead acid. Gel cells that are 12V 4Ah are very popular for alarm system back up and generally cheap. Hell, that's probably overkill and the 1.2Ah battery posted by Stoltzee will probably be enough.
But that's it! Alternator coil, rectifier/regulator, battery, you're good to go! The worst part is finding a connector for the regulator. In that case you can use a different model with a wire pigtail like this one for $9:
https://www.banggood.com/Voltage-Re...-450-2004-2009-5TG-81960-00-00-p-1077720.html
If you use the Yamaha regulator you wire the coil to white and yellow, red to battery +, and black to ground. Add a toggle switch between the battery and regulator and you're done.
Edited to add:
Here's a good candidate. It's your standard single phase 4-wire rectifier/regulator, comes with a pigtail wiring so you can easily connect to it, and best of all, it's cheap!!
https://www.amazon.com/Wingsmoto-Rectifier-Regulator-Voltage-Scooter/dp/B00ZI7U8RK
And finally, if you don't feel like modifying the Gasbike charging coil, you might be able to use a Briggs & Stratton voltage regulator from their old 5 to 9 amp "Tri-Circuit" system that used a 1-wire stator coil.
https://www.ebay.com/itm/VOLTAGE-RE...-B-S-691188-491546-793360-794360/331879044399
Connect the yellow wire to the charging coil wire, and the red wire to your battery/lights. Use the center hole to make sure the regulator is well grounded to the frame. The down side is you're actually throwing away half of the charger's output, but it will work without modifying the coil wiring. It's the difference between a half wave and full wave rectifier.