Should I remove a broken piece of the pivot rod from the trap in my bathroom pedestal sink?
You could check if the remaining piece of pivot rod that you have in hand has magnetic properties (i.e. a magnet will stick to it). If so you can probably be successful in retrieving the broken part from the trap using a tool similar to those pictured below:





Magnetic reach tools can be purchased online, hardware stores or big box outlets. In addition to the type shown above that have the flexible shaft there is another type that has the shaft made out of a soft aluminum wire that can bend slightly as the thing is pushed into the drain line. This latter type is typically a lower cost than the types pictured.