hcb3200,
Attached below is a pic of a "Dorman" replacement window regulator for your '56.
I am guessing you are referring to the center arm pivot "rivet" as being the failed part on yours???
It may be possible to replace that rivet with a thin head machine screw and nut, but unless you can find one with a shallow shoulder to ride in the arm holes, the threads on the screw will tend to chew up the hole rather quickly. Screw length must be as short as possible too so it can prevent possible interference between parts when the regulator is operated.
If you choose this path, the nut should be a self-locking style (to prevent backing off the screw from movement or vibration) or you would need to use a thread locker to keep it together.
All this being said, it may be less fuss and muss to just replace the regulator???
Am interested to hear what you decided to do and/or use to fix the problem.

Dan