My solution for the mute raising the pitch is to use a cut of loose fabric instead of sponge. By loosely running it over and under the strings there isn't the need to have it pressing against them hard enough to move them perceptibly. I use a 3" wide strip from a T shirt, and found that attaching it by string to the strap near the bridge it stays in place well enough, and then I can just remove it and shove it under the straplock until next use. Having an area on the mute so small that is making contact necessitates pressure to have much effect, and it also dictates that the mute is fairy firm, which ends up being firm enough to sharp the string when heavily muted oftentimes. I found that I much preferred my T-shirt wiper rag draped over the strings around the bridge to a foam mute, but it would shift during play. So I tied one end to the strap. Having it deliver the physical muting with less pressure from a softer material and over more coverage of the string (from the bridge to several inches up, instead of about an inch with foam underneath) and also above as well as below works well for me tonewise and vibration response-wise.
