The show-and-tell in ST4 is the handiwork of Klingons, for Klingon purposes. It thus stands fully to reason that it would feature material from Klingon sources, which explains the exterior views.
(Also, ST5 establishes that Starfleet has "virtual lens" technology so that a shuttlecraft on the surface of Sha Ka Ree can pipe up imagery that appears to be shot from a point several meters outside the shuttle. Not a major technological feat even today - it just calls for computing power.

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First, in one shot, there's a visual of the Enterprise's destruction from the point of view of someone on the Genesis planet, or at least a ship or probe close to it.
Hmmh? I don't see any such thing. The destruction views are all bow shots, which would be consistent with Kruge being in front of the ship when she starts exploding. That Kruge veers off doesn't necessarily mean he would lose the bow vantage point. (Heck, perhaps Kirk's self-destruct orders included setting the doomed ship to tail the Klingon BoP, in hopes of assuring mutual destruction!)
That Kirk's apology to the Council would be included in the material is also natural. Klingons would have access to that, as part of the process to sort out the Genesis nastiness, which is what this session supposedly is all about. The bits from Carol's report would be included in Kirk's report, so that explains them, too.
So it's the interior shots from aboard the
Enterprise that appear to be the most difficult thing to explain. Kirk pretended to be fully cooperating, even allowing the Klingons to come over using the Starfleet transporter; he could also have agreed to letting the Klingons observe
Enterprise surveillance camera material. Or the Klingons could have hacked in, like Spock did in "Balance of Terror"; it's not as if the deserted ship could have put up any resistance. That still wouldn't explain the camera angles: supposedly, the surveillance system would have dozens per each room, and Kruge choosing the exact same ones would be odd and unlikely.
In contrast, Kirk and Sarek choosing the same angles for their ST3 replay of ST2 climax is less unlikely: the material would be edited for closeups on what happens to Spock, and this is what we mostly see. The camera also briefly follows Kirk's approach, but that's also an expected or at least allowed aspect of auto-editing: the camera (in reality, the integrated and edited imagery from all the cameras) would follow any action or movement, and Kirk is the only one providing any.
Perhaps the views of the Klingons on the
Enterprise bridge come from Klingon sources, though? We know Starfleet tricorders can make visual records; Klingon instruments could be shooting the imagery we see, and sending it over to Kruge. (That Kruge seems to be relying largely on audio could be excused by his ship being shot to pieces; he can't access everything he is getting, not quite yet.)
I agree that an interesting question here is how the Klingons got access to the Klingon material. Did Kruge send it to his superiors before beaming down and perishing? This should certainly be technologically possible, but I don't see Kruge throwing away his trump card that way when he could sell it to his superiors at a high price (a promotion, a harem, whatever). At that point, Kruge is still triumphant.
Timo Saloniemi