The suggested answer, which makes sense to me, is that there's plenty of precedent for organizations to make provisions for recording of their meetings, and that therefore it would be appropriate to adopt a standing rule authorizing the Business Meeting Chair to arrange for recording of sessions at his/her discretion. However, the BM should be given the right to order recording stopped, either on a specific issue or for the entire meeting, by majority vote.
My interpretation of this would be that a motion to stop recording would be an incidental motion, but that you could raise it as a Question of Privilege when business was pending. That means it could interrupt anything except the motion to Recess or Adjourn (or some other more esoteric things), and could even interrupt a speaker; in such cases, it would be undebatable. If made while no business was pending, it would be an incidental main motion, and therefore debatable. In either case, a majority vote would pass the motion, and I think a majority would be sufficient to restart recording.
For example, if at the beginning of the first day's meeting, before any business had come up, someone could move that the entire meeting not be recorded. That would be debatable (and amendable). However, if during a contentious debate, someone moved to shut off recording (possibly interrupting a speaker to move so), the motion could be neither debated nor amended, but would instead have to be voted immediately.
I've referred the matter to the Nitpicking & Flyspecking Committee for consideration. We may come back to this year's meeting with a motion on the subject.