You could always vacuum the carpet. Did Ceasar actually paint on the canvas that was attached to the wall-carpet? Or did he just use it for display?
I have a sheet of sound board mounted on the wall that I use to attach Yupo (a type of "plastic" paper") using LARGE push pins. You need ones with a long shaft and a large head - regular push pins don't work. You need the holding power of the larger-type pins.
I question if this would work for canvas, especially larger pieces. Canvas is heavier than Yupo, and Yupo lays flat - no wringles. Canvas would require at least some tension to keep it flat. That's asking a lot from pushpins.
So i tought about cork panels, but i tried to put a sheet of paper on it and start to draw.. Its not the option im searching... I can see that the pencil, got the texture from the under level of the cork so i need something softy!
Canvas is heavy enough that you won't see the underlying texture - cork is expensive, and would have to be thick to hold large push-pins for canvas if it would work at all. I wouldn't want anything under the canvas that gives as you paint - would carpet do that?
I would try mounting a sheet of 1/2" plywood to the wall and then use a heavy-duty staple gun to attach the canvas. Whatever you use for mounting - if it's like mine - it's soon going to get splattered with paint - so that has to be something you'd be willing to look at.