FoMoCo used silver & gold paint for Mercury medallions. I think the Lucite gave it the shiny metal look. I've not tried to reproduce it but I've looked at the silver ones used on the 1951s a good bit and it doesn't appear to be any special form of paint. The 51 hood medallions use a red over silver to give that shiny look to both the colors. The gold stuff was generally for Lincoln Lido and special Mercury models like the early Monterey coupes. They just used the highest shine paint they could find.
Removing the backing plate would likely be the best way to gain access to the back of the plastic. You would likely have to destroy it to get it out without damaging the rolled edge. Find a plug button the correct size to fit back in there if possible. This is one source but there are others like McMaster Carr, etc..
https://apexfasteners.com/fasteners/...4200d7cc203d03
It might be a lot of work but those type of things are unobtainium sometimes unless you are a persistent searcher.