I have always understood the arrow with 4 veins of fletching, through an S, to be the Snider Patents trademark.
Not a proof mark.

I have half a dozen .577 Sniders and most of them have this mark. Usually the Arrow/S is surrounded by the Snider Patent wording, rather than adjacent.
Perhaps the reduced dimensions of the rook configuration caused the wording to be crowded, and did not allow space for the Arrow mark within.

NFI why the Arrow/S would appear on the butt of a Smith & Wesson Navy revolver though!

Will attempt to research the patents used in the S&W Hand-ejector, maybe one of them was a Snider patent? Seems unlikely though...