I think the biggest problem (other than having an insane customer) is that the arguments you make are weak:
- Hundreds of hours of work on a dual-screen PC will require a large amount of disk space for the recorded videos. If I don't care about space, I do care about this customer wasting my bandwidth downloading those videos.
Disk space and bandwidth really shouldn't be a concern. You'll bill both of those at a significant markup in addition to the hourly rate you already negotiated.
- Recording a video can affect the overall performance and decrease my productivity (which is not actually true, since the machine is powerful enough to record this video without performance loss, but, well, it still looks like a valid argument).
It's not a valid argument because, as you admit, it's simply not true. It may be a plausible argument, but you're trying to build trust with this client rather than undermine it, right?
- I can't always remember to turn the video recording on before starting the work, and off at the end.
Counterargument: How are you keeping track of the time that you bill? You should be marking time when you start and stop, not trying to figure out how many hours you worked after the fact. Just make the video part of your process.
- It may be a privacy concern. What if I switch to my mails when recording the video? What if, to open the directory with the files about this customers project, I first open the parent directory containing the list of all of my customers?
You should be able to handle that. Don't switch to your personal e-mail when you're supposed to be working on the project. Use an alias to get to the project.
- Such video cannot be a reliable source to track the cost of a project (I'm payed by hour), since some work is done with just a pencil and a paper (which is actually true, since I do lots of draft work without using the PC).
Your billing process should be a separate matter. If there's a requirement for video of all billed time, that should absolutely have been part of the original contract. So you're right on this point: the video is not the source of billing.
The best argument, IMO, is simply that recording every second will make you feel like you have someone watching over your shoulder all the time, and that's not something you're comfortable with. If your client doesn't trust you to work and bill in good faith, he or she should pay for what you've done and find a new contractor to finish the work (with the understanding that very few professionals would work under the required conditions).