I'm not sure how I feel about this general idea. It would be helpful if we could provide an explicit demonstration of how the formal Godel limitations apply to our reasoning about consciousness rather than making sweeping statements which may or may not apply. As it stands, I would say I'm pretty much agnostic on the issue.
In any case, we don't need to consider Godel's theorem to provide strong arguments that subjective experience cannot be accounted for by a strictly materialist model of reality. If anything, it seems to be a sort of 'cherry on top' of the standard critique. It certainly would be helpful in establishing the critique if it turns out to be a valid approach, and might even provide an illuminating basis for further theoretical results. But, again, to seriously consider it I'd like to see a more rigorous treatment. Has Penrose gone into much more detail than you have here about how the incompleteness theorem formally applies to the way we describe and think about the brain/consciousness?