While I was waiting for replacement fuses after destroying the headlight controller and blowing the battery's fuse, I thought I could perhaps power on just the headlight using a DC power supply.

Headlight opened up:

Searched for "qd139-2" (as seen on the PCB) and found lots of listings for 24-48V headlights - perfect! I poked around with a continuity meter a bit and found I could light up the light, but the JST connectors I have appear to have their polarities reversed relative to the lamp's connector!

Not ideal! But it seems to work. Also explains why things went so wrong when I hooked up the battery to the light controller - not only was connection incorrect to begin with, it was also backwards! Bad move there but oh well, learning things.