This is the first time you mentioned using AA cells to power the lights.
Please update your original diagram to include the light modules and the power to them. Also post a schematic of the light modules.
No, the problem wasn't a ground loop (although you did have one), it was noise on the ground line to the microcontroller (and hence on the output to the amplifier) caused by the LED switching current. Inserting the transformer between the output and the amplifier means that it is of little consequence if the amplifier and microcontroller grounds are at different potentials.
Inserting the transformer between the output and the amplifier means that it is of little consequence if the amplifier and microcontroller grounds are at different potentials.
But it seems that the amp instead assumes the two grounds are the same, and just connects its ground to the RCA ground. That seems like a bad assumption to make to me.
No, that's how it's done. A common mantra around here is "tie ALL grounds together" because everything needs to know what +5v is in relation to (for example.) Same goes for audio.
In your multimeter example, you're grounding the negative probe. Try measuring the voltage at the RCA output with just the red probe, leaving the black probe lying about on your workbench. That's what you get in your circuit if you don't ground the RCA terminal. So, that's why it's always done that way. :-)