How did the Klingons get the video from aboard the Enterprise?

Question

In the movie, Star Trek IV: The Voyage Home, the Klingon ambassador shows a video clip of a Klingon boarding party on the Enterprise. This was seconds before the Enterprise self-destructed.

How did the Klingons get that video?

The Enterprise was gone along with its computer banks so the video should not be accessible by anyone. And even if the ship hypothetically survived, the Federation probably would not share that video with anybody because of the sensitivities around the entire Genesis Project.

Does the novelization or any canon sources provide an answer?

Answer 1

Per the film's official novelisation, the Genesis recording was most likely taken from the Enterprise' computer records during its boarding by Klingon forces. We learn that they were able to transmit their own mission logs as well.

When their vision cleared, they saw what Sarek observed: the destruction of the Enterprise. The battered ship struggled against its death, fighting to stay in the sky, but another explosion racked it, and another, and it fell from space into atmosphere. It glowed with the friction of its speed. It burned. It disappeared in ashes and in flames.
 
Distressed, Chapel turned away.
 
How very like a human, Sarek thought, to grieve over a starship.
 
"But one fatal error can destroy the most sinister plan," Kamarag said. "The mission recordings remained in the memory of our fighting ship! Officer Maltz transmitted them to me before he, too, died. Did he die, as the Federation claims, a suicide? Or was it convenient to eliminate the last objective witness?"

_{^{Star Trek IV: The Voyage Home - Novelisation}}

Answer 2

I guess Enterprise bridge cameras would records and transmit events to Starfleet in coded message, for record keeping of events such as this. And in this case, because the ship self-destructed, that would immediately trigger the cameras to record til the last second within the possibilities.