I'm not clear if you're describing this project or a conventional freezer, but freezers/fridges are counter-productive to air conditioning because they vent the evacuated heat into the surrounding environment (your home). This is still operating on the same principle, and no matter how poorly-insulated as long as the R-value of the freezer is greater than 0 it's not going to be helping you cool your space.
I've often wondered if there's not meaningful efficiency to be gained in piping the coolant to remote coils outdoors like we do for A/Cs (with a control system to vent the heat indoors when heating is desired, or even use atmospheric cooling instead of the compressor when the outside temp is low enough).
Like a "swamp cooler". The problem is the melting happens in transit. By the time it gets to you there's very little ice and it is thus very expensive. Too expensive to leave by your bed side at night next to a fan to cool you... unless you were wealthy.
A lot of times during the really hot midday all I can think of is making ice and letting it melt during the night so it can cool the side of the bed.
If the freezer is next to your bedroom then the bad insulation becomes a good thing because you may save on air conditioning.