I know this is an old thread, but I read this and felt compelled to point out some things:
The dialog only refers to the ship's name, not it's registry number. And thanks to TNG-HD, there's evidence that a newer ship named the Constellation was in service in 2364, which is presumably the same ship mentioned in DS9.
DS9 dialog says that the Constellation NCC-1974 is still in service(never says its the "first in class", but no reason to believe it's not!)...
The dialog only refers to the ship's name, not it's registry number. And thanks to TNG-HD, there's evidence that a newer ship named the Constellation was in service in 2364, which is presumably the same ship mentioned in DS9.
While a Constellation class ship was indeed physically seen in the fleet shot, it was NOT listed on the tachyon detection grid graphic, as the Hathaway or anything else. All the other ships' classes on that list were accounted for, and IIRC none of them were Constellation class. So either there were more ships in the fleet than the graphic accounted for, or it's just a mistake. It's also not confirmed that that ship was the Hathaway. While the model did display the Hathway's registry number, it wasn't seen clearly and could have represented another ship (just like the stock footage of the Stargazer represented the Victory)....and the Hathaway NCC-2593 participated in the Tachyon detection grid during the Klingon civil war(actually seen in the episode, and on the tachyon detection grid graphic).