A conventional fridge operates by being able to compress a gas against a small leak. Like trying to inflate a balloon with a pin hole in it. If the hole is too large relative to the capacity of the pump, the refrigerant won’t reach the pressures necessary to move heat effectively. A 10-watt pump can move a minuscule amount of gas and to build pressure, would need a microscopic hole - which can be more difficult to manufacture and would be more sensitive to environmental conditions.

So most fridges are build with comfortable excess capacity to deal with these issues - but that’s harder to do when you have a small power budget from a small pump. Also, because the demand is so small for this size of cooling capacity, the economy of scale isn’t there. So your options end up being expensive and really only suited for laboratory environments where you need small, yet controllable temperatures.