UPDATE: One of my chain whips broke, so I ordered another one. The cassette has been drenched in PB Blaster since I started this thread.

New chain whip came in today, and that did the trick.

For future visitors who might have a similar question, I can confirm that the 12-tooth sprocket screws on and must be taken apart with two chain whips.

The cassette is the SunTour CS-AP20-K7 Power-Flow MicroDrive (APII), 7-speed, 12-28 tooth.

The hub is the Suntour X-1 HB-AT01-R