I put absolutely no stock in the "naval construction contract" feldercarb, especially since it doesn't fit all the other registries we've seen. It's a registry number, i.e., about the same as a car's license plate number. It has nothing to do with any kind of construction contract (why the hell would you advertise something like that anyway?).
The best I can figure, "N" stands for Federation registry, and "CC" somehow translates into the ship being in Starfleet. The Relativity's registry having "NCV" tells me that it's the first C that stands for Starfleet, with the third letter being more specific as to what type of ship it is. Maybe a Starfleet freighter would have a registry of "NCF-xxxx".
What mainly indicates to me that the Constellation is of a slightly different, and older, class than the Enterprise, is a) the registry being so much lower than the Enterprise, and b) the structural differences between the 11 footer and the AMT model. The slightly blockier 18" model works quite well as a slightly less refined design that would soon evolve into the slightly sleeker Constitution class.
This approach also allows an in for the Valiant (too old to be a Constitution, too young for a Daedelus), the Republic (NCC-1371), and, if you go with Riker's line in "Where Silence Has Lease", the first Yamato (NCC-1305).
Of course, this means the Constellation is a really old ship (around fifty years, if the loss of the Valiant is any indicator), but I can live with that.