Dawkins’s problem is that he doesn’t know the difference between probabilistic empirical theorizing and strict metaphysical demonstration, and thus misreads an attempt at the latter as if it were the former. That is not to say that Aquinas might not be mistaken at some point in the argument – though obviously I don’t think he is – but if you’re going to show that he is, you first need to understand what kind of argument he is giving, and thus what kind of mistake he’d be making if he’s made one at all.