What a find, and where did you find it!?
Hunter Arms was very aggressive in promoting their guns for trap shooting and offered a special “The L.C. Smith Trap Gun” buttplate available on any grade.
The buttplate is frequently found on late hammerless Syracuse and c. 1890-1910 No. 0 to 2 grade Fulton guns.
I think you are correct that it may be the original template by one of the L.C. Smith Maker, Syracuse engravers to be used for the mold; which would of course be a reverse image.
Unfortunately, I don't believe we have knowledge regarding the maker of the various Smith hard rubber buttplates. I've never seen one with a logo or name on the wood-side surface.
More information here
https://docs.google.com/document/d/1rtqfXbMHv08c0iojRUFUV5kR7jIc-7jdinPZEmI9WSc/edit