Full disclosure: I worked for Schwinn in Chicago from 1979 to 1984 and I had a number of Chicago Paramounts and one Waterford through the years (pre and during employment there). It's possible this came off of one of my old frames but I am really not 100% certain. I had 1973, 1976, 1979 and 1984 Paramounts both road and track.  

You can see from the photos this has the four digit date code of 3049 which I am translating as the 304^th day of 1979.  The internet tells me that day #304 in 1979 was October 31 (Happy Halloween!) and it was a Wednesday. The UAW strike in Chicago wasn’t until sometime in late 1980 so this date code is from a pretty late Chicago factory Paramount and I feel pretty good about my date code guess.  

It had been mounted to a frame at some point because the bit of wear around the screw holes is obvious but not awful. Have not tried any cleaning other than soap and water and fingertip scrubbing. It remains in very good, original condition as you can see from the photos.  There is one P-Mount head badge currently on eBay that looks like it was repainted and it's pretty scratched too but claims to be original, hmmm…. 

Would love to know how Schwinn (surely a subcontractor) had these made. Screenprinted after stamping? Pad printed? The badge is not attracted by a magnet but I don't know if it's stainless or perhaps plated brass. Really not sure. 

It would make a good replacement for a Paramount getting a recondition/new paint job. Cannot help you with the tiny screws but someone on ebay can. Finally, I will carefully tape this badge to a wood dowel and mail it in a padded envelope so it hopefully doesn't get flattened before it gets to you.  And, generous me, the First Class Mail cost to ship this 1/10 of an ounce and sort of grotesquely overpriced but pretty rare (and cheaper than others that don't look quite as good) piece is included in the price.