The video was shot when Ben was still held prisoner at The Swan.
If so then how did she know there would be an opportunity to show Jack and there would be surgery.
While Ben was being held captive, a plan was already in place to rescue Ben and then capture Jack, Kate, and Sawyer.
We can indeed assume that, knowing he had a cancer the day before the crash and that there was a gifted spinal surgeon on the plane, Ben planned to be captured and held in the Swan before he got in Rousseau's trap.
It is also important to note that Juliet's cue cards say "Ben is a liar." Traditionally mockingbirds are used to represent liars, cheats, and pranksters - which is how Juliet is attempting to portray Ben in her video. Thus one could conclude that she chose the title of the movie in order to represent her overall plot to kill Ben.
In "To Kill a Mockingbird", though, the main characters are told that to kill a mockingbird is a sin as they never harm anyone, just make music for everyone to enjoy. The mockingbird is represented by the character of Tom Robinson in the story, a black man (with the use of only one arm) who falls victim to racial prejudice.
The fact that the video was recorded with Juliet in the Others' main village means that this is part of a plan prepared before Jack, Kate, and Sawyer were taken to the Hydra station, assuming that Juliet has not been back to the Barracks since the capture.
There was a cameraman to shoot the video, since the camera zooms in. This indicates that if killing Ben is the real plan, Juliet is not the only one who wants Ben dead.
This may well have been Richard as in The Brig he gave the impression that he was one of the others that "wanted change" when talking to Locke. This was further implied when both Richard and Tom did not defend Mikhail while Locke beat him. The reason that this happened could have been that both Richard and Tom "wanted change".
The actual tape does not zoom in. The camera cuts to a close up of the Television to show only the cue cards. It is not the camera that shot Juliet's tape that is zooming in, it is the film camera that shot the episode that cuts in to a closer look of the television. So Juliet could have acted alone.
The actual tape does zoom in! In the first shot Juliet is in full body (showing the whole TV) and in the next (still showing the whole TV) she is zoomed in. Compare with the lights next to her on her right, she is not just walking towards the camera either. Although in theory she could have zoomed in with a remote. But highly unlikely. This whole thing feels more like an error.
If Karl had filmed the video then it means he knew how badly Juliet wanted Ben gone, he would've known that she's not really one of them. But, in the episode where he goes to the camp to warn the Losties about The Other's change of planes about attacking them, he accuses and exposes Juliet of being a spy, he wouldn't have done that if he had known about Juliet's real intentions.