In years past I have owned two original Sharps Carbines-however, I do not recall where all the serial numbers were stamped.
One thing I can inform you about is your statement: "There is a number on the inside surface of the patch box cover, but not the same as the serial number on the receiver..."
When the parts come off of the production line at the Sharps factory, they were placed in storage bins. Gunsmiths who were [what I call} the "Semi-Final Assembly" crew would get all the parts required to assemble a single rifle. They would put the parts together and even though the parts were manufactured on the "Interchangeable System," from time to time, there would be some "fitting" of parts required-maybe a bit of filing or whatever.
After the rifle was fully assembled, it would be taken apart, and "Assembly Numbers" [usually three or four digit numbers] would be stamped on the major parts. Then the parts would be sent out for the final finishing and hardening.
This meant that the hammer, lockplate and internal parts such as tumbler, sear, bridle, etc., would be case hardened, while the barrel would be blued and the wood parts of the stock finished with Boiled Linseed Oil.
After all of the parts gone through these processes, the gunsmiths would then use the "Assembly Numbers" to make sure that all of the correct fitting parts were re-assembled into a gun ready to be oiled and put in a shipping crate.
The use of "Assembly Numbers" was a common manufacturing process in certain industries in both the 19th and 20th Centuries.