I have an untested idea that may work for slicing a ping pong ball... Similar to the tip of the hot glue gun theory, but a bit different.
Hot wire can be used to cut foam and other similar materials, so why not a plastic ball? A simple jig could be made that would support the ping pong ball from the bottom (like an egg carton supports an egg), then take a thin wire, attached to wood dowels for hand grips, torch the sucker to make it hot, then glide it through the ping pong ball like butter.
A fairly quick downward slice motion should keep the cut pretty straight, yeilding 2 nice halfs.
Thoughts?