No.

It would be his own DNA that gave him the attribute of being able to change into a canine. He would not be a human being, he would be a different species, supposedly descended not from when we split the primates up, but it would have to be way, way back, potentially as far back as Boreoeutheria splits.

Unless you wanted to add in some fantasy mad-scientist thing, where his DNA was somehow created artificially: in which case, this answer still stands: he has his own, unique DNA.

No.

It would be his own DNA that gave him the attribute of being able to change into a canine. He would not be a human being, he would be a different species, supposedly descended not from when we split the primates up, but it would have to be way, way back, potentially as far back as Boreoeutheria splits.

Unless you wanted to add in some fantasy mad-scientist thing, where his DNA was somehow created artificially: in which case, this answer still stands: he has his own, unique DNA.

EDIT - More answer, to try to meet the intention of the story:

Further, no - If someone's DNA was altered so drastically (and it would be instantaneous, as cells are constantly dividing), it would not mean that they would have a fast transformation.

Suddenly cells are behaving differently, and there are missing cells as well that you need to create. Muscle would start growing in awkward places (breaking out bones), organ failure as they start and stop growing in areas: it would be an ugly and horrific death. Please just have one DNA for the sake of your characters.