I'm thinking about this too. The box is big enough that gluing polyurethane sheets on the inside might be OK. I think something like Gorilla Glue might be a good adhesive; it has a foaming action that would help fill gaps. You'd want to seal any gaps between foam slabs, then glass the whole inside or maybe use thin plastic to create a box to keep the water out of the foam. Seems do-able, at the expense of some ice box capacity.
I've also thought about spray foam on the outside. I bought a can of Hilty CF 116 Reusable Insulating Filler Foam that I plan to experiment with. I think it has reasonably good thermal insulating properties - thermal conductivity of 0.04 W/mK (about an R-value of 5 per inch I think). It comes with a tube to direct the spray and can be resealed and reused. I'm thinking that by extending the tube I could get the foam into the nooks and crannies at the top of the box. Another thought is drilling holes thru the box from the inside and injecting foam thru the holes. In any case, I too am churning on this.
