Hi Bob,
I assume you already fitted the box to the stock and filed the surface flush with the wood or slightly proud of it. I would engrave the box, stain the stock, and put on one or 2 coats of finish. Then reinstall the box and finish the gun. The only caveat is if you intend to glue the box in as well as use screws, the glue won't adhere to finish so either glue the box in place before finish or keep finish out of the mortise.
dave