So far, I've been able to open up the turntable kit and use wire clippers to strip out all of the internal components and excess plastic. Now the Raspberry Pi Zero W fits snuggly within the enclosure along with the PN532 NFC reader.
Finally got around to messing with the NFC reader and trying to interface it with the Raspberry Pi. Thanks to this fantastic blog post I was able to get it working in only a few minutes! Looks like the range of this chip (which is most likely a knockoff) is about 1.5 inches which is more than enough, since the "records" will be laying on top of the enclosure. Speaking of, the plastic enclosure doesn't seem to stifle the connection at all which is great. You can see a video of it in action here.