I agree that excessive compression of fiberglass insulating batts is bad - their peak effectiveness per unit thickness of space occupied may be at something more packed that the usual home between-the-studs installation, but it certainly is not when packed tightly.
Injected cure-in-place foam could be quite effective, but I would be concerned about both removability later, and distortion of the icebox due to the pressure of the expanding foam. There are low-expansion foams made to address the second problem.
My obsessive comment of the day: the stuff made of white beads pressed together is expanded polystyrene foam
, but it's not Styrofoam. Any expanded polystyrene foam can work; the extruded boards (such as real Styrofoam, or the pink stuff from another manufacturer) are better mechanically than the bead stuff (they don't fall
apart) and have a bit higher insulating value for the same thickness.