Thanks for posting the pictures of your Whitney contract musket, they help a lot. Your musket is not a musket from the 1798 Contract, however, it is one of the later contracts fulfilled by Eli Whitney, probably what collectors call the 1812 conract.
The NRA site shows a Whitney 1798 Contract Musket but also threw in a couple of images of a M1861 Navy Contract Plymouth Rifle which confuses the issue, they are photos number 4 and 5.
The musket you have, as has already been stated has been shortened. My best guess is that it as a unit was never converted to percussion since the barrel does not show any evidence of a civilian drum and nipple conversion. The lock was damaged by someone who wanted the parts for reconverting a complete Whitney and took the original hammer, frizzen and pan from the lesser condition one to do the work. Later someone attempted the restoration using the cheap cast iron pan and frizzen (both of the incorrect pattern) and a modified, possibly original to some other gun, hammer.
Can it be restored to shooting condition? Well, close-to-correct reproduction parts for the M1816 musket lock are available from The Rifle Shoppe and they can be fitted by someone with experience in doing the work. After hardening and tempering and correct screws installed where needed the lock will function, the vent of the barrel can have a liner installed and drilled and the gun will function. But will it be safe? One thing to keep in mind; Whitney's barrels had a poor reputation often showing to be "sloppily breeched" in the words of one proof master.
See a professional gunsmith for an opinion as to whether he feels it is safe, then proof carefully, don't overdo it, after all it is 200 years old. You can have the lock work done by a willing, very experienced person who has successfully done the work before, any other option will be a frustrating and expensive waste of time. Despite the hoopla over Whitney's supposedly interchangeable parts, he never achieved that goal and the parts are not drop in replacements.
Like others, I think that it looks good over the mantel. It has the possibility of being a money pit with very little reward and certainly no profit.