pseudonym58 With regard to a rug, you could first take a blanket and fold it up to roughly the size of a rug that you would want, and you may be able to get an idea of the effect on the sound it might have.
Just one thing: if the rug is going to be positioned under the pedals, you want to make sure it is also being held by at least one or two of the piano legs as well.
Testing with a blanket is a good idea. Even if the piano sounds OK without a rug or blanket it might sound better if you used one and this can only be determined by testing it both ways. Based on my experience it's not necessary to have any leg on top of the rug to keep the rug in place which I assume is your idea. This might be true if the rug is very thin and the floor is wood. In my case, the rug underneath the piano is on top of wall to wall carpeting and doesn't move at all even though the rug goes underneath the pedals and even beyond the front edge of the keyboard..