Update:
I found a work-around for this. I'm lucky in that the field that is automatically chosen as the display field (in my case, "Name") does not actually contain any useful data. Thus I can calculate this field with what I want for the KMZ.
Worth noting is that I initially tried deleting the "Name" field and adding a new field that ArcMap would select as the display field. My desired field is called "P6_NBR", so I instead went with it "NAMEP6_NBR". ArcMap looks for a field with the word "Name" in it to use as its default display field. This did not work however for the KMZ creation. Even though when I check the layer's display field post-script running it shows "NAMEP6_NBR", for whatever reason the KMZ created by my script does not use this field and instead shows [no name] or something like that.