on You Tube, look up some video's by Jim Kibler. He custom made flintlocks for a number of years and now he is offering kit guns for Southern Mountain Rifles and a Colonial-era flintlock. His video on the Colonial gun shows how he puts the gun together, including how he attached the butt plate to the stock. It could give you some ideas not only how to do it, but how to make the lines "flow" so the attachment doesn't look odd. Also, check out the American Longrifle website. A number of professional gun makers are on the site, and I've learned a lot of how these guns were made by being a long-time lurker.
I was wondering what's the length of pull on the gun in question? If it's kind of short, perhaps it was made for a child, or woman, or maybe a small man.
And what's the stock made of? Maple or Walnut? If you can match the wood, including the way the grain flows, to the original wood, you may be able to hide the extension. I would use Tite-Bond II for the glue, as it may hide the glue line better, but do some research on that.