No plastic model kit is ever going to 100% accurately portray the studio model upon which it is based. That fact doesn't automatically make the ship a different class. If anything, it would make it a sub-class of the already existing class.
The fact is, when I saw the Constellation, I saw the same ship as the Enterprise, only badly damaged, which is what I'm sure the producers of the show were going for. No effort was made to "kitbash" the model into a different design, and probably no thought was put into it's registry number.