Josh, maybe (2) "L"s for the corners (one inverted) and a "T" for the center made out of plastic to put behind the teeth to wedge it in. Just need to cut it to just the right size. That would also enable you to pull it out and clean out anything that gets wedged between the plastic and the glass sides.
Second thought, you could probably not worry about the "T" or "L" - just cut pieces of plastic the right size, put one on each corner the height of your teeth and one in the center. For the center, I would leave a gap at the bottom so it doesn't trap gunk.