Here are the facts about museums , If it already has one that is in better condition then they will not be interested in what you have.
Condition is everything on a museum piece.
Condition is everything when it comes to price as well . Is it 100% complete? Does it still function as it should meaning will it still spray paint?
While this is an old piece does not make it worth a lot.
While you may find a collector that will want it the price will not be what you think.
Plus through the history form of it Liberty and Walkup never designed that airbrush they just obtained a patent on Peeler's redesigned one after they bought his first patent.
As for your research it is semi correct
1876 - F.E. Stanley of "Stanley Steamer" fame, patents the first airbrush. It is never marketed.
1879 - Abner Peeler of Webster City, Iowa creates the first oscillating needle airbrush. He calls it a "paint distributer".
1881 - Liberty & Charles Walkup of Rockford, Illinois buy Peelers "paint distributer" airbrush patent for $700. They later pay him another $150 for 2 further refinements.
1882 - Peeler's patent for the "paint distributer" is granted on April 25th with Liberty and Charles as assignees(owners).
1882 - Liberty Walkup applies for a patent on an improved version using Peeler's new "walking bar." It's granted on September 18, 1883.