So what I did was measure the voltage at the connector that the incandescent holster plugs into. There are 3 wires there and I found 12V between two of them, so I fashioned an electrical connector from some thick wire I had and inserted it into the 2 appropriate holes in the connector that is running inside the scooter, apparently to the controller.
That worked, but I wasn't really bright enough, so I got a second LED light and mounted it in front of the dash. I wanted to connect it in parallel with the existing make-shift connector and this is where it started to get a bit messy as I didn't have everything I needed to make the connection. I tried wrapping the new LED's wires around the appropriate part of the thick wires that I had inserted into the connector. The second LED lit up too which was great, but I was in a bit of a rush and didn't secure the electrical connection very well.
Later that evening both LED's went out and when I measure the voltage at the connector at home, there was now no voltage across them. I suspect that during the ride, some of the wires must have short circuited and burned something out in the controller.
A little while later, my bike started behaving very strangely. It turns on with a key just like a normal scooter. However, my bike was turning on by itself even when the key wasn't inserted. When the bike was on, it would not respond to the throttle, but periodically jerk and stop, jerk and stop. When I tried to push the bike home, it was somehow applying something akin to a regenerative brake, even though my bike doesn't have regenerative braking. So it was basically impossible to push my bike anywhere (because of this magnetic braking effect) and it wasn't responding to the throttle. I was stuck for a while wondering what I should do.
I was lucky that I had taken off some of the plastic covering the insides of the bike, that I hadn't bothered to put back yet (as I'm still tinkering inside the bike), and I could just reach in and disconnect the battery. After doing that, I could push the bike. Then gingerly reconnecting my battery, I found that the bike's controller had reset to a sane state, akin to rebooting my computer (controller) and I could ride the bike home.
I was quite concerned about the bike after that and I went down and checked on it several times that evening and into the night with a volt meter. Luckily, it wasn't doing anything bad to my battery, but the burn-out of the connector for the lamp stayed dead. The rest of the bike seems to be functioning fine though after the reset.
Well, I learned that the light on my bike is connected to the controller and short-circuiting probably did something weird to my controller, putting it in a kind of unexpected state. Resetting the controller by disconnecting power seems to put it back into a sane state. Right now my plan is to run a wire from my battery through a separate switch to the lights taking the controller out of the control loop, and to use proper XT60 connectors instead.