Going from First Contact and Enterprise, the idea that Cochrane lived in Alpha Centauri in his later years seems a bit improbable. He was still working on the Warp Five project based on Earth after all. When Kirk is saying, "Cochrane, of Alpha Centauri", could he not mean "Alpha Centauri project", a name for the Warp Five project with a goal to reach Alpha Centauri? That would make the line make more sense with later canon. The way you might say a professor is "of Notre Dame" and mean Notre Dame University, and not the recently destroyed cathedral. Does this retcon possibly work?