Honestly, There's precedent for the ship just being rededicated. (although it happened well after the fact and due to budget issues in the CGI department) The Defiant NX-74205 was destroyed, and a couple episodes later a new Defiant class, the Sao-Paulo, was rededicated to be the Defiant, even gave the ship the Defiant's registry. No reason that one of the first couple of Constitutions couldn't have been rededicated Constellation, with the Registry NCC-1071. If the previous Constellation did something spectacular, like maybe sacrifice itself at Axanar to save the ship that would later become the Constellation while it was being built, would that be worthy of a re-dedication?
The whole thing with the Defiant's registry was of course, they were just about out of money in the CGI department and just used old stock photos of the old Defiant whenever they had to show her on the screen.