After some catalog-hunting, I settled on the TLC5947 for my LED drivers. It's an SPI-controlled PWM LED driver with 24 channels. I'll need two of them, but I'd like to get a feel for how they work before I commit to a design.
Here's a $10 technical test, with an Adafruit-designed breakout and minimal OSHPark boards. It's using the same side-mount LEDs in the same orientation as the final design, and I'm pretty happy with them.
Only issue is that I have no idea what colour each package of LEDs is, they're labelled in Chinese!
For anyone that wants to make the test boards, my "miscellaneous" PCB repo is here:
It costs $6.10 for three.