If TSA finds out that watermelon and oranges can help you
hyderate yourself.. they will be considered as WMD and would be
banned soon. Better be ready to get a prescription for those.

On an important note, I second your idea of the bottle with filteration.
There is nothing in the TSA manual that will ban you from taking it on the
flight, and if someone stops you, you can contest it. You should also make
it clear to the airline when you book the flight that you require extra
water due to a medical condition and they should have extra bottles on
board for you. If anybody gives you trouble, you can always mention the
fine legal system of the United States.
(I'd rather not take things that way, but mentioning it can get things
done)