The Franklin baseball from target has a core of cork, with no rubber around the cork core, so it is the correct kind of material for the abuse you are about to put it through. First, take your baseball and remove both the leather covering and the yarn, you should be left with a sticky ball of cork. You can try to make it less sticky; I just ignored it since I was planning on sealing it later. You do want to sand the cork down a little bit and clean up the worst of the rough edges.
Next, using your band saw (or jig or maybe even a chop saw), you want to cut off the "top" of the ball so you have a flat surface to work with. Make sure your cut is pretty shallow. You can then make a very shallow cut (like just scraping it a little) on the "bottom" of the ball; it will make your life easier to have two flat surfaces.
Take your 1 1/2'' spade bit and drill out the center of the cork. You need to drill down until you are just at the point of breaking through. You will need to go low and slow on the speed; it's pretty easy to shatter the ball by going too fast.
Switch to the 3/16'' bit. You are now going to drill the holes for the lights to poke through. I did two rows with the holes spaced approximately one inch apart horizontally and then vertically. Do what you think looks correct. Once all your holes are completed (and you have checked to make sure the lights will enter the hole), spray the ball with plasti dip and let dry. Repeat this step as needed to insure you get a fairly smooth surface.
The lights are in the image because we were testing them - you can do this step but be aware that you will have to remove the lights before the final assembly.