I have gone through and soldered connectors in place so the device can be powered with an external 5v supply. I added a speaker to the top of the box as well as a pushbutton. The results can be seen below.
This has worked okay, but the audio has been very quiet. I realized that the sound level decreased when I removed the speakers from their original cases.
Looking at the specs for the LM386, I realized that I am at best getting less than 325 mW of output power through the speakers because I am driving the amplifier with a 5V supply. I went ahead and bought a couple of 5V, 3W amplifier boards on Amazon. I added this board between the output of the original board and the speaker. In this process, I increased the software volume to 100% while I reduced the LM386 gain to 20x from 200x. The volume is now substantially louder and is relatively clear. I am still encountering a bit of clipping at the highest amplification settings.
The internal setup of the box is seen below. Components have been double-side taped into place.
As an alternate option to using the Raspberry Pi Zero, I ordered an MP3 module. This arrived and I tested it. It works well and I began integrating it with an Arduino Mini so that I can still just play 1 song with a single start/stop button. In the process, I damaged the module; so I am going to be continuing with the Raspberry Pi Zero for the time being.