I think someone saw an oportunity to put together an RJK gun.

With an 'of the period' engraved M12 of what appears to be of quality and in my opinion a set of wood from another gun (perhaps an orphaned stock set), spliced together, then the RJK signiture added afterwards thru the bluing finish to add still more value.

The pheasent doesn't look quite up the 'K' standards of lifelike quality to me, the dog looks good. But again it's hard to tell from small pics.

The near 100% metal finish and extreme wear to the wood,,especially the grip checkering, still makes me think someone more recently put the two componenets together rather than they were originaly built that way.
Perhaps they are original to the gun and the metal has been reblued w/o polishing. Then an afterthought to add the RJK signiture.
A few different ways to go with it.

Seems like RJK would have at least marked it originally if it was his. He wasn't shy about signing his work, even if it was just a floorplate for a rifle.
Lots of engravers from that part of Europe cut that style of pattern,,they apprenticed at the same place. There's even some around today that can cut the style if they want to. It's not like it's a lost art or something.