Build it out of what though? You might as well make whatever way you can to a store which has those portable iceboxes I vaguely remember seeing in my first exploration of the city. Pull water from the pool out back, fill the icebox up a small amount and freeze the water, then lay the food on top of the ice. It'd be a simple matter of making certain the ice remained frozen then. If you insist on saving it anyway.

[He should be capable of turning a small amount of water into ice at least, so that he and Kimblee could possibly take turns at such a task if the two decided to pursue it.]