This is a large "pinback" button that measures three and a half inches across. It was made by Banning Enterprises (who also produced a unique licensed poster that year) in 1985, and has a solid metal back.
It will be interesting to see if any further items from Banning Enterprises turn up; it wouldn't surprise me at all if there were more out there, especially considering the [normal-sized compared to today] merchandise push for GODZILLA 1985. Here is a higher-resolution scan of the artwork!
12/13/2021 UPDATE: There I was, reading my own post from 2018, about the Banning Ent. poster mentioned above, and there it was: a flyer allowing you to join the Godzilla Fan Club, which, in doing so, promised you would receive newsletters and.....a limited-edition button! Is this mystery solved? I am betting that the above pinback is the very same one!