See if you can scrounge around for some thin sheets of styrofoam, say 1/4 to 1/2 inch thick that you can line the sides of the bin with. Otherwise, use a towel. Or use a towel in the meantime. I had this problem with one of those small refrigerators I purchased just for plants, bulbs etc. Except the whole inside of the fridge would freeze even on the warmest setting. I put a sheet of styrofoam just under the freezer and that seemed to do the trick.
I also think wood would work just as well.
