I unscrewed the six screws holding the speaker closed to reveal a 4 ohm 20 watt speaker, a 4 ohm 10 watt speaker, and a small controll board for the sensor.

I desoldered the 3.3µF capacitor and speakers. I tested the capacitor for a short, it was fine. I measured both speaker's and their resistance did not indicate any short. With the speakers desoldered, I plugged the speakers back in and started the tree. The lights did not dim this time. I took this to mean that the sensor control board was not causing it, but to be sure, I resoldered the speakers and desoldered the signal (NON), 5 volts (VCC), and ground (GND) wires from the sensor control board and turned the tree on again. This time the lights dimmed, and the audio got garbled after a while. The issue seems like it has to do with the sound system or power supply.