I will try to add some additional information to help you make sense of all of the different model numbers that have been thrown at you. It is a confusing topic, and not all collectors agree, nor does the Ordnance Department correspondence always remain consistent.
The production of Muskets had not yet achieved interchangeability, thus a musket made at Springfield would not necessarily match a musket made at Harper's Ferry. The way that contracts were done was through models supplied to the contractor. In this case, the term model refers to an actual physical musket that was provided to the contractor by one of the armories.
The contract would read like this: "The Muskets to be manufactured, shall in all their parts, conform in model, size, and form, to the pattern or model Musket, furnished by the Ordnance Department, and shall be equal in workmanship and quality, in every respect, to the Arms made at the National Armories during the same period."
Notice that the contract does not actually specify what the Model name is supposed to be. In the case of the Musket you are getting, it was manufactured under a contract issued on Jan. 26th, 1829 for 8,750 Muskets and bayonets to be delivered over five years to Watervleit Arsenal in New York.
Thus, you would not be incorrect to call it a Pomeroy Model 1829. Most collectors, however, would call it a Model 1816. Some would call it a Model 1816, Type III. Schmidt calls it an 1822/28. The Ordnance Department referred to it as a Model 1816, but differentiated between Muskets made before or after 1822. They later issued manuals that referred to the Model 1822, even though they didn't seem to call them that at the time they were being made. In other words, it's all pretty muddled.
Just my opinion, but for your presentation, I would refer to it as a Model 1816, made under a contract issued in 1829. Hope that helps.