I believe that particular line of Brunswick balls has the numbers, stripes and stars applied with a silk-screen type process and sits on the outer surface of the balls - very thin and will rub off quite easily if too much pressure / heat / abrasive is used.
IMO, I'd ONLY use the Aramith brand Ball Cleaner / Polish by hand of course. On that particular cue ball, apply the cleaner solution directly to the ball with your fingertip in a circular motion as if you were massaging every square inch. Then grab a clean microfiber towel and buff gently. Then repeat once.
With the object balls, massage the cleaner gently - and only on those areas that you detect with a naked eye - that really needs cleaning. Then buff them gently with a clean microfiber towel. Once is enough on the object balls.