Among barreled meat (defined as at least 196 pounds in each container), clear pork was the best class, comprised of the sides of large hogs, with the ribs cut out. It was destined for New England residents who "in the line of pickled pork, buy nothing short of the best." Mess pork came next in quality (and price), as it included two rumps as well as the sides. The navy and commercial marine took a great deal of these. A barrel of prime pork was one step down, typically contained sides from lighter hogs along with two shoulders and two jowls. It was marketed for maritime "ship use" as well as "the southern market" where it generally ended up on plantations.
mess: a building or room in which members of the armed forces take their meals; mess hall : the sergeants' mess.
Apparently, therefore, mess-meats are lower quality cuts of meat intended for consumption by the masses (including ship's crews and so probably in army messes too).