I finally got the Gotek drive flashed with the FlashFloppy firmware: https://github.com/keirf/FlashFloppy
Unfortunately, I ordered the "wrong" board which requires the ST-Link to program it. Fortunately, I happened to have an ST-Link here. However, the drive is not quite working as planned. I think since it's not a very supported board by the community, I may have something backwards. Changing the jumper over to a different set of pins from what it recommended, the ST detects drive B and tries to read from it, but it comes up as garbage in the directory.