[PaleHawkDown]

Maybe a zip gun?
I wonder how accurate the drawing was to the original. I collect old newspapers, and in the era before photography, there is often a bit of creative license.

 

[PaleHawkDown]

That image seems to show a plunger on the rear of the gun, like a cocking piece. I bet if you pull that out, the trigger moves forward.

The tubes on top are a mystery.

At first I thought that nubbin on top might be a sight. It looks a little like the rear sight on an 1871 Colt.

It sort of looks like there is a hammer or something behind the top rear tube. Maybe there was a hinged middle section to this that was secured to the "sight" and the sight was really a stud? Or maybe the "stud" is part of a hinge, and this was a breech loader on the top. Maybe a pistol/shotgun combo?

Maybe the big tube on the bottom is being used as a frame housing the trigger mechanism, and the top is the only thing that fired, but we're missing part of it.

Or, here's the long shot guess, the whole thing is a primitive lower receiver, and we're missing the upper.

 

[PaleHawkDown]

Those top tubes are sights aren’t they?

That's what I thought at first, but in the second picture it looks like there is something behind them. Also, what is the "bump" on top of the tube. A decent photograph, and a different angle would help a lot, right?

 

[PaleHawkDown]

If that was lost in 1812 and not found until nearly 100 years later, little would be left of the wooden stock. I don’t see any means for a trigger action to be in this gun as pictured. Considering the drawings, it is most likely that this is a toy gun that some kid playing cowboys and Indians lost fairly close to the time that is was discovered.

The OP said it was found at that site, but it could have been made any time between then and when it was found.

You can see the trigger in both pictures. It's one of this style of trigger:

These started popping up in the late 1840s. The grip shape started popping up in the 1850s.

The gun, if gun it be, is clearly not from 1812, but we're just playing here and throwing ideas against the wall.

If we assume it IS a gun, the bird's head grip means that the earliest would be 1850s.

 

[PaleHawkDown]

I think that you are on the right path. How about this air gun
View attachment 184002

The Pope Brothers Rifle Air Pistol, called a rifle because of its long barrel and detachable wire shoulder stock, was manufactured in the USA circa 1874 to 1878 by the Pope brothers (Albert and his younger brother, Arthur) at the Pope Manufacturing Company, 45 High Street, Boston, Massachusetts, USA.

The handgrip is made of metal.

I think that is it exactly. Even has the right sight, trigger, and grip.

 

[PaleHawkDown]

This is absolutely the best forum. Our mystery solvers would give the Scooby Doo gang a run for their money.

 

[PaleHawkDown]

I wonder if the original writers in 1905 ever figured this out, or if this forum was the first to do so?