Hi Jeremy and welcome. Here's what I would do:
1- check for blow fuses in the converter.
2- wiggle the light
bulb in the socket. If it lights you have a dirty contact.
3- work the switch on and off to see if the lamp lights. If you get a flicker you have a dirty switch. I'd use contact cleaner on that.
4-pull the fixture looking for a bad connection.
5- check the connections at the converter.
With multiple light
not working my bet it's 1 or 5
From this point it gets more involved . Good luck, Raz