Unfortunately, serial numbers were seldom, if ever recorded by the Ordnance Dept., when the weapons were issued, so that information is unusual, and fortunate. The other thing is that these arms were not necessarily
shipped in the same crate with others in the same number range. However, the fact that your rifle is only three numbers away from the one that is listed could be an indication it was
in the same box. The gap in years between 1884 when the rifle was manufactured, and its being issued to the 5th Infantry in 1889 means both rifles could have been almost anywhere in the intervening years.
Often, if there was an individual mentioned as having an arm issued to them, it was either the post ordnance sergeant or officer, or a commanding officer of some kind. You might check with the National Archives to see who "P. Kane" might be. Try for a military record, and also see if there might be a pension record.
Many times, even if the person's military record doesn't give much information, if they lived long enough to claim a pension, they often wrote detailed accounts to back up their claim for the pension!
In addition to trying to track the name, is there any mention of the unit to which it was issued? If you can find that, it may help find what post the unit was assigned to. You might want to also check to see if that individual might have been an Army Quartermaster Dept. employee, rather than a soldier. Some times these individuals "lost" or purchased the arm and had the money deducted from his pay. You might then try to track when that company (arms were usually issued by company), and to see when that outfit was issued Trapdoor Springfields. Unfortunately, the ordnance records are not known to exist after June of 1876!
If they were, no one at the Archives seems to know where they might be.
Caution: Do NOT try to shake these rifles or otherwise try to make them talk!
It would be great if they could!