The proof checker checks proofs within some formal theory. The Godel sentence for the checker is certainly true and provable by us, given the consistency of that formal theory. (If the theory were inconsistent, the checker would be able to prove any sentence.) But this doesn't work as a proof within the theory! The theory cannot believe its own consistency (Godel's second incompleteness theorem), so the checker cannot assume it when checking proofs. So your argument doesn't actually give an example of a valid proof rejected by the checker.