I think you're beginning to learn a painful lesson on why you don't produce a PCB in large numbers until you've had a chance to run it through the wringer for a while. ;-)
I agree with you that the noise is probably not caused by a ground loop. Trying to alleviate the noise using techniques intended to fix ground loops is just going to be a waste of time. There were some good suggestions a while back on trying to isolate the high current loops using capacitance and such. Did you try that? I know it's not what you want to hear based on board layout, but if you can, test the theory anyway. If for no other reason than to prove it wrong and move on to something else. Do whatever you have to do to get a large cap on the power rails near your LEDs and see what happens.
You have a large pulsed current flowing from the battery to the LED boards, via the microcontroller board. Because of the resistance and inductance of the power supply lines, this means that the ground on the microcontroller board carries a small pulsed voltage relative to the battery negative terminal.
The transformer fixes the problem because you feed the difference between line output and microprocessor ground to the primary (thereby cancelling out that small pulsed voltage) and this gets reflected at the secondary.
1. You really should provide adequate power supply bypassing on the LED boards. Without this, your boards may also generate RFI. I think you need at least 1000uF between +5V (the positive supply to the LEDs) and the ground pin of the TLC5947. 2200uF or 4700uF would be better. An inductor in series with the incoming +5V on the LED board (as discussed earlier) would be the next step.
2. Include a small PCB-mounting audio isolation transformer on the microcontroller board.
3. In the diagram in your original post, you have a ground connection to R11, R12 and JP7. If you choose not to include a transformer, then make provision on your board to separate this ground from the main ground on the board, with a jumper connecting them together. When using a common supply, remove the jumper. This way, the voltage divider will be referenced to the amplifier input ground instead of the microcontroller ground, so at the same time as reducing the signal voltage, the divider will reduce the noise voltage passed to the amplifier. Removing the jumper will eliminate the ground loop too.
4. Use thick wires between the microcontroller board and the battery (to reduce resistance), and ensure the power and ground conductors are in the same cable, or twisted around each other (to minimize the area they enclose and thereby minimize the inductance).
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?
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?