The main concern is, the fridge is designed to chill air inside, not really solid elements.
What you must consider is chilling power per cubic centimeter of air. Say, a 70W chilling power applied to 1 liter of air wont make the same result with 1000 liters.
Now put a hot surface in that chilled air. The contact area will be cooled down either by 70W/l or 0.07W/l to put it simply. Of course you see, the smaller the volume the better.
What i mean is , you should try to reduce the air volume inside the fridge. Maybe ducting fresh air straight out of the compressor onto hot surfaces is the key.
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