Another option that you could consider for a one time problem like this is as follows.
Work at getting as much of the trap dry as possible. This could be done by using your mechanical grabber to push dry strips of rag down the drain and pulling them out again.
Once the trap has most of its water removed you could attempt to further dry the inside of the drain by forcing air flow down the drain. A warmer air will help dry the trap more than cold air. I would caution against using really hot air, such as from a hair dryer, because this could damage the plastic components of the drain system. Also note that you want to avoid sucking air out of the drain into the room due to the smell and stink involved with that.
Next make up a loop of wire made from a thin coat hanger that you can easily push down into the trap area. Using the loop arrangement lets you have the option to twist the end around better than just a single wire. It also gives more "size" to the end of the wire assembly. (If you find the steel coat hanger wire too stiff you can also try bare copper electrical wire of 12 or 14 AWG).
Finally, attach some pieces of Gorilla Tape to the loop end of the wire with the sticky side out. The idea is to get the sticky tape down into the trap to capture the broken piece of the pivot rod and then pull it back out with the wire. Do make sure to attach the tape to the wire loop in a way that even if it tries to stick to the inside of the drain pipe you are able to get it back out without leaving it down in the drain pipe.
Gorilla tape is an incredibly sticky tape with a tough fabric back that can be purchased at hardware stores or big box outlets such as Lowes or Home Depot. It is fairly expensive but once you have a roll I assure you that you will find other uses for it.