[Dave McCann]

My idea for a jig would be to screw the piece to a small board (I used a screw chuck to turn the piece) and attach that board perpendicularly to another board (with the ball resting on that board).

Similar to this? (which may need support block under the "ball" portion depending on the diameter of the base portion of the chess piece).

As I understand it, this fixture above, is only the 1st half of an entire "jig". This fixture does nothing more than hold the work piece for free hand cutting on the bandsaw.
If you are making multiples and want repetitive results, there needs to be a way to index the fixture angle to angle and maintain a known relationship to the bandsaw blade.
Thus your work holding fixture could be set on top of a sled base, that is guided using the miter slot in the bandsaw table. Dowels or stop blocks in the sled will locate the holding fixture in either position "a" or position "b" for repeatable results.

Red line indicates blade path.
Tan rectangle is guide rail on bottom surface of the sled to engage miter slot in saw table.