Here's a quote that I think you may find interesting, as it seems to be along the lines of what you are saying:

"A meta-mathematical proof of the consistency of arithmetic is not excluded by...Goedel's analysis. In point of fact, meta-mathematical proofs of the consistency of arithmetic have been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936. But such proofs are in a sense pointless if, as can be demonstrated, they employ rules of inference whose own consistency is as much open to doubt as is the formal consistency of arithmetic itself. Thus, Gentzen used the so-called "principle of transfinite mathematical induction" in his proof. But the principle in effect stipulates that a formula is derivable from an infinite class of premises. Its use therefore requires the employment of nonfinitistic meta - mathematical notions, and so raises once more the question which Hilbert's original program was intended to resolve." -Ernest Nagel and James Newman

Taken from https://www.mathpages.com/home/kmath347/kmath347.htm

Lucas himself foresaw the 'expand the axioms' response, and pointed out that once you do this, you get a new formal system with a new Gödel sentence. This means that no such argument against his thesis - i.e. by adding the truth of the formal system's Gödel sentence to that system's axioms - will terminate.

Note that I am not saying the Lucas-Penrose argument is indefeasible; I am saying that it cannot be defeated simply by the argument that goes 1) it claims the human mind is consistent; 2) we can see that the human mind is not consistent; ergo, it is false. This argument fails because premise 1 is false. If it were this simple, the video here would not be nearly two hours long!

Addendum: The "stepping out" argument is not a matter of adding axioms, it is a sort of meta-argument that a Gödel sentence for a consistent system must be true[1], and the consistency of the hypothetical system under discussion follows via the Church-Turing thesis (which you could challenge if you want to, but I suspect that, to most people who understand the implications of doing so, this would be an undesirable trade of one implausibility for another.)

This leaves me wondering if the Lucas-Penrose argument can be challenged on the grounds that the algorithm in question probably does not halt (other than for causes outside the system; namely, the causes of death or the causes of a power outage), or that the question of its halting is undecidable. I am sure this response has been debated somewhere, if it is not obviously wrong.

[1] https://math.stackexchange.com/questions/1491061/paradoxhow-...