When I saw the title, I thought of a paint bucket lid I shot up once. It wasn't self-healing, but it wasn't fragmenting like other plastics either. Pellets tended to push through, leaving most of the plastic merely deformed. That made of LDPE (most are HDPE). You can buy LDPE lids at Lowe's, but certain food storage containers are made of LDPE too.
Then I read the post and saw the requirement for "quiet". Hmm. LDPE isn't a steel gong, but it would make some noise. My next thought was maybe a polyurethane casting resin or latex mold material from a place like Hobby Lobby. Of course, that might start costing some money and maybe make those ones that Active Self Protection promotes an option after all.