The spring is necessary to balance the upward force of the fixture's internal mechanism; otherwise the servo stalls out when it tries to pull the chain down. The spring is hooked to a piece of 3mm plywood that I cut out when I was prototyping. The plywood could easily be replaced with a 3D printed part, though.

The chain is attached to the servo arm using a bead chain connector heatset into the PLA. I also friction welded some PLA around and through it with a Dremel, and since then it has stayed put. This allows the light chain to be easily detached from the puller. 

The puller seems to work on the first try about 70% of the time. I have added some code to move the servo part of the way before moving it the whole way, and that seems to help it. If the problem persists or the servo breaks, I may try using an MG90 servo (with metal instead of plastic gears) for higher torque and better durability.