Zem: you could do that. what you would need to do though is get a large container. Maybe even use a 5gal bucket and make a shelf that would hold it. Have that and little else in the fridge as it works off air circulation. You would then just route 2 hoses to that bucket. One hose would be a return and the other would connect to your circulating pump.
Then you would need to get about 4'-6' copper tube that will fit just inside the hose(I used 1/2" copper tube) so that it can be clamped tightly onto the copper tube. before you connect anything, you would gently make small bends in your tube (which will probably come in a coil already. Don't get copper pipe, that has thicker wall and won't bend easy). You want to form a coil with the copper tube so that it will fit inside your rez and will have both ends facing the same direction (don't get too fast when bending or you will crimp it). this will set in the rez, submerged in solution, with the two hoses connected to it.
This will keep your nute solution and chiller water separate, while the copper tube will allow the heat to transfer to the cold water that you will circulate continuously. This will work and it won't run the fridge to death(a small fridge may not be able to keep up but not sure). I used to have a chiller that worked on the same principal until LEO decided to take it. I used a nice, well insulated cooler for my cold water reservoir, for you the fridge with the 5gal bucket will be your cold water reservoir(you must have the cold water reservoir to prevent the fridge from running constantly and it will cool the Rez solution better. As long as you can stay between 65-72f in your rez then yer golden.