not sure about this. but if the fridge is new enough the condenser coils are built into the sides of the unit rather on the back like they used to be. so side clearance could be an issue????4" on each side is nonsense lawyer speak for CYA purposes. You need back ventilation for the compressor heat to escape and you'll be more than fine. You could downsize the unit also and get clearance all around if you (or the wife) is hung up on the 4" thing, but it might look funny if it's got gaps all around. Like a motorcycle parked in a warehouse.
Also, if you're just swapping for a Samsclub mini and you have no ventilation at all, you can replace it every 2-3 years and still come out ahead.Or you could cut out slits and ventilate it yourself. Unless it's rock. I wouldn't want to mess with cutting rocks.
Congrats on the new house!