Another dirty little secret is the newer TLC5947. It has an internal oscillator so you don't need to worry about the 4096 clock BLANK interval, and it also has 24 channels so you can control exactly 8 RGB LEDs with one chip. It cannot handle higher current like the TLC5940, but it is enough for normal LEDs.
I guess for your application you could either have 12 LEDs on each side and make them meet in the middle, or have 24 LEDs on each side and have opposing pairs connected in series. I don't know how big your model is, seems like a really nice solution would be a PCB milled the internal shape of your phaser window area, and side-looking surface mount LEDs.