Interesting...it sounds like it might be tricky to get this to work. But, theoretically not impossible, especially if some company would manufacture a unit specifically designed to work like this.
I have been thinking that if you had the need to run a chest freezer anyway (I have one because of my large garden and infrequent shopping trips), you would have a way to get by without a fridge. My idea is you buy one of these well-insulated 5-day coolers (non-electric), like this: http://www.4x4xplor.com/coleman-xtreme.html
. You keep some bottles of water in the freezer, and put them into the cooler when you need to keep something cold. You rotate the water bottles as needed to keep it cold. Now, this would give you only limited space for fridge-stuff, but it might be enough if you changed how you do things. For example, a lot of the stuff in my fridge is storage (flax seeds, for example) that could be vaccuum packed and stored at room temp instead. During times of the year when there's more fresh produce, just add another cooler!
Recently we've been eating a lot of watermelon and sweet corn. When I bring these home they are too bulky to fit into my fridge, so I have set up a (regular style) cooler in the kitchen where I deposit the watermelon and bag of sweet corn. I pop in a couple of frozen water bottles and this keeps the produce nicely chilled for at least a day, often two days--once we've carved up the watermelon and eaten some of it, I chop up the rest and put it in a container in the fridge.
BTW, we have all these plastic drinking bottles but have switched to drinking out of stainless steel ones. I'm not throwing the plastic ones out, though--they make great freezer bottles. Fill just 80% full to leave room for expansion. You won't be drinking this water, just re-freezing it, so the plastic doesn't matter. They fit easily into crannies in the cooler and take a long time to melt, lasting much longer than ice cubes.