The debate in my mind is this... sure it's important to see the barbaric nature of the ISIS group, but on the opposite site, you know there are people in the world who get off seeing this kind of stuff. So the issue is one of "do we show the truth, in an effort to raise sympathy for the victim, when it will inevitably become a source of enjoyment for some, and a source of sympathy for the enemy for others.
In my mind, I think you can show enough video to prove that it really happened, such that it raises the outrage of those who's outrage Fox wishes to raise, without showing the entire 22 minutes of it. At some point, it stops being evidence of a crime, and instead becomes an orgy for masochists.
To express this another way, imagine you were called to sit on a jury where a man was accused of producing thousands of images of child porn. As a jury member, would you want to be forced to watch EVERY SINGLE image/video, or would you at some point say enough is enough, I don't need to see that to know the guy is guilty? In some cases, it is actually necessary to do so, to ensure that justice is done. But the court certainly isn't going to release all those images to the media to convince the court of public opinion that the guy is truly guilty.
Similarly, someone in the Japanese government should probably see the whole thing just to confirm that it wasn't a fake, and it really happened, but at that point, what purpose does it serve to show it in it's entirety? None really, except to gratify those who either support ISIS or get off on this kind of thing. If knowing it happened, and seeing a short clip of it isn't proof enough to stir up the national outrage to finally put a stop to it, no amount of video will.