CO2 levels varies greatly depending on the temperature of your beer. To carb to normal levels, youd need around 30psi if your beer sits at room temps but only 10psi at fridge temps. I would think if you are having to constantly swtich out ice to keep the beer cool, this could be a hassle to get consistent results. I guess what im trying to say it the portable kegerators are best used for that purpose, not as a long term kegerator.
That being said, you could do it, just may be more work than its worth. The carb levels should equalize when the beer cools down but youll have to constantly adjust the regulator pressure to compensate