As Abomination said, I'd go with getting a mounting kit, routing 1/16 of an inch (the thickness of the mounting plate) under the art.
As to "all of the ball and only the ball showing", that's harder. The trackballs have that black "lip" around the ball; this is designed to be level with the top of the CP. If you cover the lip, especially with the art and plexi over it, you are going to "lose" that much of the ball.
If, OTOH, you have the "high lip", aka "golden tee" TB, it is designed to have the 1/16" mounting plate + 1/8" plexi over it, with the top of the lip level with the top of the plexi. You'll have a black "tube" around the ball sticking through the plexi. With the lip around the ball, you won't need to round the plexi at all, just sand the jaggies (if the 3" hole in the plexi is perfectly round).
The third option is if you have happs "high ball" model (IMO, yuck). This model can have the lip covered with art + plexi and have enough ball over the plexi not to feel like it's buried, but IMO the ball jumps off the rollers too easily for me to use effectively. (Others love this ball, though.)
Here's an old image showing how much ball would show above the 1/8" plexi: