Thank you for your question to History Hub! Based upon the information you provided above and the information available online, your rifle is not an 1878 model, as Springfield did not have a model of that year number. Rather, it is likely your rifle is a Springfield Trap Door Rifle of the 1873 model vintage, with a particularly deep date stamp which makes the 3 appear to be an 8.
As a knowledgeable source on the subject states of later-produced Trap Door models: “Even though the guns were different from their parents, they were still marked “Model 1873.” It might be noted that many Model 73s had very deep stampings, which made the “3” look like an “8,” leading some collectors to think there actually was a Model 1878 Trapdoor.” See link.
In addition, given the serial number you provided, the National Park Service records of the Springfield Armory site narrow the production timing of your rifle to between January and June of 1875, based on the order of production available by serial number.
Beyond those ascertainable pieces of information related to the production process of your firearm, the history of your particular gun would likely be difficult to substantiate without the existence of records documenting the specific serviceman(men) who used it. It is possible additional documentation of this specific rifle is in the custody of the National Park Service, though you indicate above that you have already contacted them. It might be worthwhile to follow up with the Springfield Armory site for more information based on the specifics above here.
The National Archives also has records germane to your search on microfilm in Record Group 156 (Records of the Office of the Chief of Ordnance). These eight rolls can be made available to you in the microfilm room of the National Archives building in downtown Washington D.C. However, keep in mind that this entry (Summary Statements of Quarterly Returns of Ordnance and Ordnance Stores on Hand in Regular and Volunteer Army Organizations, 1862-1867, 1870-1876) only substantiates the existence of ordnance on hand and likely does not go into detail regarding how those arms were distributed in the field. However, if you are curious about looking into this microfilm further for references to your weapon, contact their reference staff at firstname.lastname@example.org. This article from NARA’s Prologue magazine also further details the archival research process pertaining to firearms and their provenance.
Hope this helps and best wishes on your continued research!