From the Dossiers on the official Web Site, she's a Survey Vessel, and that would fit with the dialogue heard at the start of the film.
She's absolutely great, and I loved her introduction at the beginning of the movie.
She has probably around 800 officers and crew, and enough shuttles for an evacuation, or other missions as necessary.
Her crew would make sense, as well as the countless shuttles, if we consider that she may need to make extensive observations of planetary environments etc., where a large number of survey officers go to different parts of a planet, making detailed and thorough observations very quickly.
The single Warp Engine also makes sense, since speed is not really much of a concern, but the capability to perform large studies is (hence size and crew).
In the Prime Timeline, after this point, the Constitution Class is more a multimission ship, and far more automated in many areas (such as Engineering), meaning she scould be built smaller.
In the new timeline, however, the destruction of the Kelvin, and the fact that the Romulans turned up, resulted in a delay in the Constitution Class project, different Engineers being used, and a shift in the way Starfleet viewed the technology which led to a later built, slightly more advanced, and much larger Enterprise.
But yeah, I'd love to know more about this ship in some "Canon" form. Even if we know her official Class name would be great.
Personally, I'd like to propose a Class Name for the man who headed the team that gave us this movie, and IMHO, rescued Star Trek: The Kelvin should be designates as an Abrams Class Survey Ship.
Of all the things I've lost, I miss my mind the most.