I did think of this. The whole punch needs to be big enough to obscure the exact location of the dot but not so big all the information trying pass gets lost. My solution was to repeat whatever message or image multiple times on a single page in different locations. That way if the message gets whole punched it shows up in another location so everything is still readable.

That's just the price of avoiding tracking. Do the procedure, create the hole-punched document and complain to the printer manufacturer about the result :)

Well you would need the whole punch big enough that the exact location of a dot would be obscured. Like, every hole would be not perfectly centred over a dot. If it was centred you could essentially establish what the dot location was prior to cutting and give away what you are trying to not do. So what you suggest would show an approximate location of a dot but not reveal the details.