This isn't a British P1796LC. It's American, a type often called an "1812 cavalry sword" or similar. Late in the 18th century, growing tension between the USA and France, followed by, early in the 19th century, growing tension between the USA and Britain (which would lead to war in 1812), the US government and state governments proceeded to build up their armed forces. One of the results was this type of sword, with Federal contracts for 500 swords in 1810 (which are sometimes called the M1810), and larger contracts with the State of Maryland in 1812 and 1814. These swords typically had iron guards. However, many similar swords, often with brass guards, were made for sale to various militias - the Militia Act of 1808 made money available to arm/re-arm militias. If your sword is unmarked, it is likely to be a militia sword. If it is marked (but I see no markings in your photos), it would be Federal or State.

The blades on the swords made in this time period vary. Some are much like the British 1796, with a wide fuller (but often without the widening near the tip of the 1796, and often with a clip point rather than the hatchet point of the 1796). Some have a narrow fuller, like yours. Some have flat blades without fullers. The narrow-fullered blades appear be a design originated by James Winner, who was the original contractor for the 500 swords in the 1810 contract (but couldn't make them all, so the contract went to Abraham Nippes, who made most of them).

For discussion of these swords, see Andrews, Jacque, Philadelphia Gunmakers and the Evolution of the "Maryland Sword", Bulletin of the American Society of Arms Collectors 89, 10-16 (2004): https://americansocietyofarmscollectors.org/articles/18/

The classic reference book covering these is Richard H. Bezdek, Swords And Sword Makers Of The War Of 1812, Paladin Press, 1997.

For plenty of various examples, and a lively discussion, see http://www.swordforum.com/forums/showthread.php?118038-What-Did-I-Buy