I have a few pin badges, that would just sit in a box.
I couldn't find anything I liked that was suitable or simple enough to display them, so I drew up these three versions.

These were designed to fit Dia 26 o/d pins.

For my print I also added holes for the pin location, but have removed these as I found that not all the pins were in the same place.

I would suggest making a paper template of your badges pin position and then transfer it to the recess. Then drill out a hole suitable size. I also had to use a thin piece of rubber sheet, as one of my clasps wouldn't clamp the pin enough to be a tight fit.