Another possibility?
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Hmmmmm … That does look interesting.
Earlier tonight I decided to take the plunge though and, cutting out a small rectangle of cardboard out of an old banker’s box lid, I used double-side adhesive 3M tape to attach the backside of the badge to the middle of it - fashioning sort of a press die like used in old-fashioned newspaper printing. I bent up the edges of the cardboard on each end to fashion little “lift handles”. Taking then a (waxed) paper plate, I sprayed a good-sized spot of the center of the top side of it with my 2x indoor-outdoor silver spray paint that I mentioned earlier. While the paint was still wet, I quickly and carefully lowered the badge face down into it by the lift handles, where I pressed the face hard and evenly into the plate. Carefully lifting the assembly then by the handles straight up, I was surprised to see how well an initial coating of the silver paint applied to the face of the badge letters, leaving the black-painted edging untouched. Only problem was that the paint kinda “stuck” in certain areas of the face when I withdrew the “die” from the plate, so I was left with a rough-looking surface rather than smooth if I had used a spray gun instead. I tried then pressing hard (and carefully) downward again with the die but against a small piece of clean waxed paper instead of the plate. That “flattened” and smoothed out the paint on the letters, but still it did not end up looking as smooth as it would if sprayed on. I waited 1/2 hour for the paint to dry, and then repeated the process for a second coat. It now looks a little better than after the 1st coat, but I did get some silver paint on a couple of tiny areas of the black edges.
I am going to let things set up fully overnight and tomorrow night I will try the process one more time (3rd coat). Assuming the paint still won’t be perfectly smooth, I’ll use my tiny model paint brush to try to smooth it out as well as can be before it sets up (hopefully no brush marks!). I’ll let it then sit another 24 hours to dry and then touch up the spots where the silver got onto the black edges with another brush using my original flat black spray paint. After that has thoroughly dried, I will then, as someone suggested, give the badge a final 2 coats of (lacquer? polyurethane?) clear coat to protect it and to give the silver hopefully more of a “shine”, and I then should be done. 
Maybe you could use the existing badge to create a plaster mold, and use the plaster mold to cast a new badge out of pewter or some such.
If you’re curious, there are several YouTube videos on this - creating them in Aluminum and Brass. - Molds are often made of Silicone - OEM part removed… mold injected with wax… - sprued to a wax tree… coated in investment (plaster like stuff) … wax melted out… brass/Alu poured in…
~Paul K.