I don't know whether Dawkins would agree with the following version, but here's a reworking of his ultimate 747 gambit into what I think is a good argument that the Abrahamic God probably does not exist. The argument is not meant to be a knock-down one. If we had some reason to believe in disembodied spirits or in substance dualism, or if we could rely on some other source such as revelation, then the argument could be defeated. But in the absence of a convincing independent argument for God's existence, and on the assumption that what we've been finding so far is that we really do live in a disenchanted universe (with no ghosts, spirits, metaphysical dualism and so on), I think it's a cogent argument. It also has the merit that it doesn't assume that metaphysical naturalism is true; it merely assumes the much weaker (and I think highly plausible) claim that we've never yet encountered anything like a spirit ... something capable of intelligence, design, and so on, but with no complexity to it.
The argument won't work with everyone, but it should confirm to a non-believer why the likely non-existence of the Abrahamic God follows from the rest of her worldview (without that worldview having to presume fully-fledged metaphysical naturalism). It should also be successful with a science-oriented religious doubter who is coming to think that we actually do live in a disenchanted universe. If the Universe is, as it were, disenchanted internally, that's a factor in how (un)likely it is that something like God exists.
Here goes, for comments.
Whenever we look at something that has actually been designed, as opposed to something that has evolved over millions of years by natural selection, we always find that the designer is an incredibly complex being. We know of nothing in the universe that packs more complexity than the human brain, and we know of no process more complex than the brain's functioning as it moves from one massively internally-connected physical state to another. We simply have no experience of anything that is known to have been designed except by a highly complex designer.
Thus, if by analogy with objects such as watches and telescopes, we inferred that the Universe itself is designed, we would also infer that the designer is something incredibly complex. To make any other assumption would be arbitrary.
Once we reach this point, we must ask where the incredibly complex designer came from. The only ways that we have ever observed by which massive complexity comes about are by design or by the simple iterative process of natural selection over vast spans of time (and needing vast volumes of space for there to be conditions amenable for it to happen).
Hence, it is highly likely that the designer of the Universe, if there is one, was either itself designed by something else OR something that evolved. It is inconceivable that such a thing just sprang into existence.
However, both of these possibilities are incompatible with the supposed nature of the Abrahamic God. It follows that all the evidence available to us leads us inexorably to the following (very) likely conclusion: either the Universe is not the product of a designer (in which case the Abrahamic God does not exist, because this being is said to be the designer of the universe) OR the Universe was designed by something that does not match the description of the Abrahamic God (in which case, again, the Abrahamic God, as per theological descriptions, does not exist).
Therefore, it is probable that the Abrahamic God does not exist.
Abrahamic theists are likely to reply that their God is not only the designer and creator of the Universe but also a being that is simple, e.g. with no moving parts, internal linkages, or changing states.
This does seem to be a logically possible state of affairs. However, we have never encountered any entity remotely like this. E.g., we have not encountered the "spirits" that some Abrahamic theists talk about. To claim that God is like something that, in turn, is of a kind we've never actually experienced is a desperate move.
Once again, if we had some independent basis to believe in the existence of such things as spirits, then we'd have a basis to infer that the designer of the universe might be a being like that. But as we've come to know more and more about the Universe, and have failed to encounter disembodied, simple intelligences such as spirits, we find ourselves in a position where we have no basis at all to conclude that the universe as a whole was designed by something like that.
Note that this argument is only probabilistic. It does not say that there cannot be a God like the Abrahamic one - that it is just impossible.
The argument is "merely" that if we could think about the idea of design, without being prejudiced by our familiarity with religious ideas, and if in doing so we relied on our actual experience of the world around us, we would reach the conclusion that the Abrahamic God probably does not exist. If we could think about it clearly, with minds free of prejudice from familiarity with religious ideas, we would conclude that the probability of this being's existence is actually extremely low.
In short, the existence of something like the Abrahamic God is a bare theoretical possibility that's contrary to all the data that we have so far from our actual experience.