I bought 5 Meters of IKEA RGB light two weeks ago and I kinda want to embed this into the project. The problem is: the Pi only has two PWM channels and I'll have to use both for the system fans, ignoring that 3 are needed anyway.

The Texas Instruments LP5009 and LP5012 seem promising, they're essentially made for this. I haven't dug that deep yet, but from what I've read so far, you could control the driver via I²C. Obviously, the power button I had planned would interfere with it (it requires the SCL pin of I²C).
But I can either just ignore the button or come up with a "path switching" solution, where it still uses the I²C-To-GND short to wake it from halting, but then uses a different pin for the power off (or I'll only include a hardware-wake and no hardware-shutdown).

I'll have to check how things work in reality, maybe I'll find a different solution or maybe I can keep the button there, since it will only be used once for waking it up. (Speaking about that: I'll have to check if I can use Wake-on-LAN instead and completely remove a physical button).