Mandolins and banjos use what are called loop end strings. These strings end in loops and the loops fit over pins. So you should be OK, if you can get loop end strings, to fit them onto the pins.

What I call ball ends are approximately 3.5 mm diameter and 3 mm deep, and yes they have a groove around the perimeter. If the measurements match, my ball ends and your fat washers may well be the same thing.