The hanging weigh idea may work but it will take considerable weight. Laying flat with weight has not been as effective for me as I find with most materials, leather, cloth, even thin metal a material needs to be bent beyond the desired shape before it will spring back to where to want it. I often coax the leather into the shape I desire by over bending a bit but no so far that you kink the leather surface away from the deeper layers and completely ruin it.

You can wash the cloth. If cotton it may or may not get a little stiff but will soften again. Linen will get rock hard when wet but will also soften again once dry. In either case it will shrink a bit and will wind up shorter than your leather section.