It's a relic. Leave it as a relic.
All 1816s were made with manually reproduced parts, so very little, if anything, is interchangeable. It's not worth looking for an original stock to butcher it to fit these parts. IMO there isn't a reason to put it in a new stock either. They aren't cheap themselves and will require a significant amount of work and skill to make it passable. Even then I don't think many folks would believe that doing so would improve its displayability.
Barn finds like this are getting hard to find as complete muskets since the sum of the parts is usually worth more than the gun. They've been busted up for parts for decades, so it is nice to see a survivor. I know a couple of guys who collect stuff just like this for that reason.