I think it may not need one, as the original never had one and they never got that far in the build to have a badge made for the special. Jimmy never mentioned any badge at all.
However, they did make their own coaster brake hub, so the brake arm definitely needs a name stamped on it.
I found some interesting data on coaster hubs, they came out around the time the Wright's were making theirs.
"The inventor of the coaster brake in the USA has been the subject of much speculation. During the 1890s many different versions of coaster brakes were used in bicycles and most of them were never patented. During this era, modifications to the coaster brake were quite common. As an example, the Wright Brothers developed a coaster brake for use on their bicycles but they never applied for a patent. This surely was not entirely their invention, but instead, it probably was a variation of an existing design custom-made for their bikes."
They usually used a frame built by other companies for their bikes, like Pope. That looks like a Pope head badge on that bike in the shop:
"Both the Van Cleve and the St. Clair bicycle were largely assembled from parts made elsewhere. Frames, handlebars, seats, cranks, and tires were purchased from sources such as the Davis Sewing Machine Company of Dayton, OH (which later became the Huffy Corporation), Sager Manufacturing Company of Rochester, NY, and Pope Manufacturing of Hartford, CT. The only parts the Wrights were known to have made from scratch were their bicycle wheel hubs."
I could easily get a blank cut out that looks better than the Chinese made arm, or just have that one stamped. I've looked to see what the arm looked like, but there aren't any good closeups of a Wright hub that show the arm.