Well, I got the cooling unit out when I got home this evening and got a good look at the bottom of the chest. It is in worse shape than I thought. The rust was just falling apart when I would touch it, so I went ahead and knocked off a lot of the loose stuff. Looks like a new bottom of the chest is in the works.
There is a hole under the rust where insulation used to be. I think some spray form insulation should work for that though.
These pics are before I knocked off a lot of the loose rust.