I was listening to Subspace Radio's interview with Cpatain Geko, where he explains the reason why the Constitution didn't show up in the fleet ship list. The ship is considered kind of sacred in the IP, and doesn't even show in later series like Deep Space Nine during battles involving other old IP ships.
But what about that gorgeous refit design in the game, the Exeter-class? I always loved the design of that model and surely it doesn't carry the same dogma as the Constitution it was based on. When I saw the game's promo video when it went free to play, the shot of the Exeter was the highlight for me, a sign that the ship designs I had grown up with as a kid had been modernized faithfully.