Mike,
Use caution in using distilled water. Topping off is fine. Does your tap water necessitate it?
I understand reasoning behind it, but the fish won't live in it. If you want to make the argument that the chlorine kills the good bacteria, ok, but the lack of minerals in the water will kill the fish. Regarding chlorine, ever see a frog live in a pool? (high ppm chlorine/sensitive animal) They live for quite a while.
I haven't read anything about an under gravel filter yet.
Have you considered an experiment with an empty jug and tap water? Let it sit for the same time as the aquarium and see if the cloudiness you describe occurs. It'll help you know where the problem lies.