New boards arrived and I soldered them all together. I discovered the SOT32 footprint of the battery charging IC and the micro USB jack were both really tough to solder. I bridged a few contacts a few times and had trouble fixing them. Flux certainly helped fix things.
The lights show up WAY better than they did with the CR2032s. Before, the lights were really only visible when the lights were dim, now they're visible all the time. (Note: this picture is with a bright light shining right on the board.)
There are two downsides to the current setup:
First, I had to make the whole thing bigger to accommodate the battery, which makes for a rather large name tag. I think I might add a second magnet so the name tag doesn't tilt downwards.
Second, the battery life isn't great, on the order of a couple of hours. I might be able to increase the resistance in series with the LEDs to eek a little more battery life out of it, but I really want to just add more juice so I can keep it nice and bright.
Next up is a more generic board so I can use this places other than the makerspace.