Which has no bearing. The number system is formal and contains Godel statements. It is (assumption) the natural product of human reasoning which is (another assumption) a natural product of physical law. By composition, the number system is a natural product of physical law. Ergo, physical law contains Godel statements.
Human reasoning is, I maintain, complete, in that all possible truths are in principle available to it ...
Not so. Human reasoning has bounded algorithmic complexity. Any truth exceeding that complexity is not open to it. So far as I know, there's no way out.
... including the truth or falisity of formally undecidable Gödel statements
Again not so. The nature of Godel statements is that they are neither true nor false but may be assumed to be either true or false.
I don't think I need to make a case that human reasoning is inconsistent.
Actually I think "unreliable" is what you're after. Inconsistency would be believing X and not-X at the same time and not thinking that's a problem.
Is it known that there are provable theorems out there that are hopelessly beyond the human intellect to prove? I've never heard that claim made before in quite the way you just made it. Of course, there are a whole host of things that are undecidable in Godel's sense of it, but that's a fundamental constraint having nothing to do with our intelligence or lack of it. I've often thought that perhaps there is some "critical mass" ito intelligence, and that once a species exceeds that critical mass, it can in principle prove anything that can be proven or, alternatively, understand any valid proof. As Albert Einstein was supposed to have said, "The most incomprehensible thing about the universe is that it is comprehensible." Hope that's true. It's gonna be depressing if we're just another species of dumb apes.
But by the same argument, human reasoning would contain Gödel statements, which is patently absurd. Gödel statements are not a general feature of mathematical systems. They only occur in certain classes of formal axiom sets, such as number theory.
Human reasoning has bounded algorithmic complexity.
Show me. Remember that we are talking about human reasoning in principle, so don't bother with in-practice arguments like the limited human lifespan.
The nature of Godel statements is that they are neither true nor false
No, the nature of Gödel statements is that their truth or falsity is not provable by the set of axioms of the formal system in which they appear.