As I recall, there is no in-universe confirmation that the dog actually died.There's no suggestion in the movie that Scotty's instructor (relativistic physics?) was Archer, and there's thin proof (canon-wise) that the Admiral Archer spoken of was still alive.And that Archer was still alive, and teaching at the Academy, in Star Trek (2009).
"It is," didn't appear on screen at any point, so no "it isn't."
Yes! +1 for in-universe interpretation!
(See Star Trek novelization for the CBS-approved conclusion to the transported-dog story.)