@ J.T.B.
I don't need to assume what the first two registry digits stand for, because the creator of the Enterprise, Walter Matt Jefferies made that obvious in his TOS pre-production sketch, IMHO:
http://www.trekcore.com/specials/albums/sketches/STTOS_Sketch_EnterpriseConcept17Cruiser.jpg
And for the context of TOS and into the first films it works perfectly (Constellation notwithstanding, plausible rationalizations possible):
17th design series - Enterprise Class
18th design series - Miranda Class
19h design series - Soyuz Class
20th design series - Excelsior Class
That was obviously written before anyone knew that "NCC plus number" would be applied to all types of Starfleet ships. You bring in the movies as support, but the movies prove it wrong: The NCC 1700s can't mean the 17th cruiser design because a 500 was a scout (Revere) and a 600 was a science vessel (Grissom).
Was "J-class" shown on any display? If not, maybe Mendez was using a common 'nickname' for the class. Perhaps it was a "Jester class", or a "Jackal class," or a "Juan-Carlos class," or a "Juniper class," or a "Jiffy-Pop class," etc.....
Or it could be a class of ships whose names all began with J like the British J-class destroyers (Jackal, Jaguar, Javelin, Juno etc).