How would we know it was a specific make/model bomb that exploded? I doubt there'd be a "W88 Copy" label thrown clear and found in the rubble. As to identifying the radiation fingerprint, if anything that would point the finger at China, presuming we know the particulars about their particular flavor of plutonium. They stole the design for the weapon, but they didn't steal fissionable material from us.
We would know from the massive yield that a W88 produces (500kt) compared to a backpack nuke (under 1 kt) or a Pakistani nuke (40 kt)