I haven't done a project like this, but if I did, I would try some plastic/rubber material. I just bought a piece to line the bottom of a shelf I made for my kegerator. It is sold bulk, by the foot, and there are several different styles. I would get a rough cut, then use a new single bladed razor or utility knife to cut it exactly. (Overlap the ends slightly, then cut through both layers for a perfect fit. Then trim the top and bottom flush with the lid using the razor.) I would try a spray adhesive on both surfaces. Thats what I would do. Good luck and attach some pic's when its done.