Once yo've determined that the side of the fridge is not a heat exchanger, you could cover it with a "stainless steel backsplash"
This would insulate by reflection.
To further prevent heat conduction through the metal to the fridge, you could back it with plywood. The plywood does not have to be thick, and 1/4in might do. You can get sheets in full size (4x8ft) but also in half or quarter size.
You can glue the backsplash to the plywood with a high temperature adhesive (plenty available), and I would further support it with screws in the four corners, in case the glue fails. Depending on the size of the backsplash and its rigidity, the glue may even not be necessary and several screws in/around corners would suffice. The screws should be short to fasten the metal to the plywood, but not pierce out the back.
The plywood can be hung from the fridge with L brackets or a cleat resting on the top of the fridge. Here you could use double sided tape to prevent it from sliding off, as the actual weight is held by the shape of the bracket and not the tape.
Should the fridge require clearance, you can space the plywood from the fridge using standoffs like washers or small pieces of cut-off plywood, to allow air to move between plywood and fridge.
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