No, they can't. Its not a matter of what universe to put it in, its a matter of they legally can't use anything excpet already agreed upon story elements (the destruction of Romulus) from the new movie without getting sued by Paramount.
Anything else and they would have to go to CBS, get them to agree to go to Paramount, then hammer out a new deal. It could happen eventually, but that's something that is probably years off, if ever.
I suppose we'll have to settle for a unique single nacelle starship, then. Or maybe something like the Freedom class.
As much as Franz Joseph's book is derided, I'd like to see some of the ships listed there in the game. Saladin-class destroyers, Hermes-class scouts, Federation-class dreadnoughts, and for those that want to make a living hauling, Ptolemy-class transports.
Proxima class from ST Legacy game. Also Posiedon class from same game as costume option for Miranda/Centaur/the other one ship.
USS Kelvin (since it existed prior to the Romulan's temporal travel it would exist in this timeline as well)
TOS Miranda from ST Legacy game... at least I think thats one of the ships in it and not a mod, I don't recall
Star Trek Starship Creator ship parts for appropriate class vessels (ie soverign model pieces for soverign... Oberth for Oberth)
New Orleans as seen in TNG, though that was all damaged up though.
ST movie style Constitution class style, saying (This design was one of the initial designs of the Constitution class but was rejected in favor of the one we all know and love by Admiral ______, who is noted as being Captain of the USS Kelvin) which would explain why that model was used in the movie because that character died and couldn't reject it. Also those other ships that went with the Enterprise were most likely older and probably would have been built the same way in the standard timeline.