
You can easily switch this to 12V AC, have brighter lights, and stop worrying about battery maintenance.
First, remove the battery.

Get rid of the useless regulator/rectifier that was the reason your 6V battery wasn't charging.

Open up your headset and ditch the "sealed beam" 6V headlamp that doesn't have a replaceable bulb. Swap your connector ends with regular spade terminals for the replacement bulb housing.

Install the new bulb style headlamp, swap out your 6V high beam and speedo bulbs if you want. I ordered the wrong style, so I let it ride and they didn't blow later when I tested it, so I think they might live.

Swap your brake light for a 12V version:

Mount up a 3 pole 12V PX style regulator, the hole pattern is the same so it's no sweat. Make yourself a ground lead for the black pole on the regulator.

Shrink wrap the ends of the white that feeds the turn signals and the white from the stator.

I decided to use a 12V DC Lambretta junction block to mate the power and loom leads. If you buy the DC version, all of the terminals except the green pair are electrically connected, so you can just plug in your regulator connection to one and the loom leads for the headlamp and brakelamp circuits anywhere else (except the greens, of course). Take one of the yellow stator wires and give it a loop and ground it on the regulator, along with your ground pigtail you created and with the loom ground. The other yellow goes to your new regulator. Make a yellow lead to connect the regulator output to the junction block. Tuck the red wires back into the loom sleeve to keep it neat.

Shrink wrap the white ends and the extra ground terminal. Tie it all back neatly and tuck it out of the way.

Now you are ready to rock bright 12V lights. If you want, order a 6V AC horn to complete the conversion. There is no 12V AC horn but the SIP catalog says the 6V will do fine. I ordered one but mine hasn't yet arrived.
Want to make this conversion easier? Buy all of the bullet connectors (3.5mm), spade connectors (F63 and F63S sleeve), and a Professional Kit CK-1 which includes a quality crimping tool from vintageconnections.com. He ships fast and the stuff he sells is good quality.