idea
Checkable or uncheckable mechanics
If the mechanics are checkable (physics is computable and the brain follows fixed rules), the theorem has something to grab. Any rule-following mind would have true statements about itself that it cannot establish from inside, and it would take its own soundness on faith.
If they are uncheckable, say consciousness depends on something non-computable (Penrose’s bet), the theorem does not apply. Either way someone still has to show that a mind’s reasoning contains arithmetic in the relevant sense.