Since all official, Paramount-licensed publications refer to the ship as Constitution class, I myself see very little reason why it is not that class. But that's not what the OP wants to know.
I wasn't even aware that Jeffries built the Constellation. If he did, then I can't think of any logical reason why he wouldn't have been just fine with any registry number other than the one he put on the model, unless he felt that something like "1710" would look too much like "1701" from a distance. But really, why would that even have been a concern? One ship is new and pristine; the other is a destroyed hulk. Regardless of the registry number on the saucer, no one was going to mistake one ship for the other.