#moustache {
  position: absolute;
  bottom: 10px;
  margin: 0 auto;
}

If that doesn't work:

#moustache {
  position: absolute;
  bottom: 10px;
  left: 50%;
  transform: translateX(-50%);
}

Relative positioning is preferred but not always available if the parent face is positioned absolutely.

Edit: adjusted bottom from 0 -> 10px since 0 would be at the bottom of the chin but there is obviously some padding to bring it nearer the lip