scswift:
I don't understand. Is it because the ground plane and wires act kind of like a capacitor as power flows in and the battery is unable to discharge them back to 0V quickly enough that the ground potential of the board fluctuates as the LED modules draw power?
No, as he said, the ground traces act like an inductor. They're unable to change instantly. With DC pulses, the voltage (in theory) goes from 0v to +5v instantaneously, then after the PWM delay, back to 0v instantaneously. In real life, that just can't happen. But, the fatter and shorter those traces are, the closer to the ideal behavior you will get.
scswift:
So am I right in assuming that the reason putting a cap in series with the RCA's ground pin didn't work, was because the fluctuations on the ground were AC in nature, and could pass right through the cap?
No, my last answer to this still stands. You formed a very high-frequency lowpass filter. You probably eliminated the MHz-region noise adequately, but it takes big caps to filter low frequencies. Of course, if the noise is already in the audio band on the RCA cables, a low enough filter will remove audio as well. That's why the proper fix is to prevent the LF (audible range) noise from leaving the source of that noise at all.
I sympathize with your quandary. Prototyping electronics is often an expensive process, but it has to be done. Real-world constraints affect performance in big ways. Data sheets and example schematics are never enough to tell you what is really going to happen. Out of a couple thousand dollars spent at fab houses, I'm not sure I have even a single project where the first PCB was ready for mass distribution. I am nowhere near confident enough in my engineering ability to even consider promising a product before I have a completely-built prototype that I'm able to put hands and scopes on. Although, sometimes I can hack it or live with it enough to use it, but I still go back and touch-up the design just in case I ever have spare room on a future PCB order.