I think the argument in the Third Meditation is valid; at least "mostly valid", meaning with a few minor adjustments, or perhaps some implied premises (obviously something can't be literally "mostly valid". In a nutshell, and with a bit of simplification, it goes:
P1) My idea of God has objective reality, X
Descartes makes a distinction between the amount of 'Objective reality' and 'formal reality' that ideas have. The amount of formal reality a thing
has can be thought of as how complex that thing is. So the Colosseum has more formal reality than a stone, but the idea of a stone
and the idea of the Colosseum
both have the same amount of formal reality, because the things we are concerned with are both ideas. However, ideas have content; they represent, or are about, something. The amount of objective reality an idea has depends on the complexity of the content of the thought. Thus the idea of the Colosseum has more objective reality than the idea of a stone.
P2) For any idea with objective (or formal) reality X, there must be a cause with formal reality X or greater formal reality.
The Causal Adequacy Principle. It seems to me to concern both causation, and the origin of ideas. Concerning causation, an appropriate analogy would be "For any object to have X amount of energy, there must have been a cause with X amount of energy, or more", i.e. cause and effect involves some kind of transmission of properties from cause to effect, which I believe has traditionally been a popular view on causation (before Hume, that is). Concerning the origin of ideas, the implication is if we have an idea (a clear and distinct perception as Descartes would put it) of something complex, then there is some extra-mental thing that is at least as complex as that.
P3) I do not have formal reality X or greater
Descartes bases that on the idea of God as an infinite and perfect being.
C) God exists
My point is that it is very abstract metaphysical beliefs (the Causal Adequacy Principle, and Descartes' notions of existence), rather than particular 'counter-evidential' beliefs, that get Descartes into trouble. The question that follows seems to be, can metaphysical beliefs be 'counter-evidential' in the same way that (I think, because I haven't read him) Flew is talking about? They can certainly be out of line with how we ordinarily use metaphysical concepts, but I think it is difficult to be so damning as to say the metaphysics of certain philosophers is counter-evidential.
That said, there are lots of arguments in philosophy that we don't class as sophistry, but that we immediately feel are somehow fallacious, yet can't quite pinpoint why straight away. I think Descartes' Third Meditation falls into this class, and early-modern philosophy in general seems to be full of that sort of thing. For example, Descartes' two proofs of God, Berkeley's Master Argument, perhaps even Hume's arguments on induction and causality (although, there are at least three or four plausible interpretations of Hume's views on causality, and one might feel different about each). Are these kinds of arguments bad? I don't think they constitute bad philosophy, because they tend to nudge along our progress and understanding rather than hold it back. Indeed, the example you give about immaterialism hints at the problem of consciousness, which is at the forefront of the philosophy of mind today.