I have been re-reading the extract from Berkeley to try and pick out some of the flaws in his arguments.
B. states that to reject the principles and theorems of science is not equivalent to denying the existence of a material world, since the theorems of science are "intellectual notions, and consequently independent of matter". B. has misunderstood what it means to deny a theorem. It is not the same to deny that theorems exist as it is to deny the conclusion of a theorem. The theorems of science posit things about the natural (real or external) world and to deny them is exactly to deny the existence of the external world. B. cannot claim the "common sense" position without absurdity.
When talking about secondary qualities, B. moves from the argument that "To exist is one thing, and to be perceived is another" to the position that whatever we perceive cannot be the same as what exists (in the case of different temperatures being felt by two hands and in the case of pain) and from this to the rejection of material things existing at all. What B. is missing is the explanation that things appear differently at different times and under different circumstances. All B. has proved by the hands in water example is that we have a point of view of reality.
B. uses the example of the same objects appearing to be different sizes at the same time to different animals as an argument to reject primary qualities as existing in objects. I don't really follow B.'s argument unless he is suggesting that for something to exist it must appear exactly the same under every point of view, which is the argument he used for secondary qualities and is false.
B.'s mis-step then is to knock down the straw man that he puts up, but the straw man is not a good one. In order for there to be an external world it is not necessary that existence is exactly the same as perception.