As a past software manager of a elite team(of super smart people, engineers, phDs) and developer myself:
Never do that, say NO.
Don't try to justify it. Don't lose your time with this person, don't lose emotional energy.
It is one thing to be efficient, another to "look efficient". You can be not efficient but look like you are or the contrary, specially when your client don't have a clue in software.
The reason this customer wants the video is to go against you, as you said, he already does not trust you.
When you do, he is going to go and say: look, you are not working 80% of the time, so I am paying you 20%. Probably he already have plans for doing that.
He probably does not have the money, or the organization changed plans and don't need the software as much as when the agreement was done and wants to backpedal with any excuse.
Don't try to justify it. Don't lose your time with this person, don't lose emotional energy.
It is one thing to be efficient, another to "look efficient". You can be not efficient but look like you are or the contrary, specially when your client don't have a clue in software.
The reason this customer wants the video is to go against you, as you said, he already does not trust you.
When you do, he is going to go and say: look, you are not working 80% of the time, so I am paying you 20%. Probably he already have plans for doing that.
He probably does not have the money, or the organization changed plans and don't need the software as much as when the agreement was done and wants to backpedal with any excuse.