Poor Professor Z! 

A kind official at the Utah club says that they use Acrylic paint.  Therefore, it seems that an acrylic silver paint for touch ups, and then an Acrylic finish (aka clear coat) should be compatible. It seems like a good idea to get a regular Professor Z that you don't care about to run a test first.  Get a good, blue, masking tape, or paint-on masking material, from a hobby shop to mask PZ's plastic parts and wheels.  The directions will probably say to spray the finish in many fine coats.  You also may want to get a very fine sandpaper (the highest number available at your hardware store or hobby shop) to lightly sand before applying any finish, and then between each coat of finish.  You are going to build up fine layers of finish in order to achieve a smooth coat.  Try to find a well ventilated room with clean air to do this.  If you do it outside you will probably get a lot of particles stuck to your finish.

If it doesn't seem to work then you could try a different brand.

I'm thinking that your previous botched attempt where the finish rubbed off could have been finish that didn't match the paint, that the finish didn't adhere to the smooth, unsanded, factory finish, or that the finish wasn't allowed to completely cure (IE you have to let it dry, AND harden fully). 

Please correct me on any of this anyone, as I am surely not an expert.

Good luck, Alicia!