YAY! Budget increase.
Now I have enough to manufacture my own PCB!
I won't be driving the LEDs in parallel anymore, I'm going to use 12x 500mA drivers at 350mA and then split the PWM signals from the arduino. I'm currently in the process of drawing up the circuit, I'll post it here when I am done.
EDIT: Is it okay to use a 350mA driver to drive 350mA? I know it's always a good idea to underrate electronics but I can't find a chip that does exactly what I need it to for with more current. I need an IC that can handle 24V input, 350mA output with a PWM frequency range that supports both the 500Hz and 1kHz speeds on the arduino pins.
It seems that the driver I listed in my original post is my best bet. Their datasheet seems to point to the fact that it was designed for use at 350mA.
Besides, this unit isn't going to see more than a thousand hours of use anyways. Not that I'd want to make it un-reliable, but if I reduce the IC'd MTBF to 40,000 hours from 80,000 it isn't a huge deal.