(Disclaimer: All of what follows is explicitly done under the assumption that the mind is a Turing machine and that we can formally axiomize our mathematical thinking. This has, of course, never been shown to be possible. What I'm describing is the state of affairs that would obtain if that were possible, given the metamathematical and metalogial results of recursion theory and set theory. More is said about the plausibility/nature of this assumption at the end.)
The question of whether or not the human mind is a Turing machine (or to be more specific, Turing complete modulo unlimited memory) has been around since the 1930s. A lot of ink has been spilt on that question as well as the question of what consequences either answer has for the philosophy of mind. There is a well known argument, the Lucas-Penrose argument, that the human mind cannot be a machine because if it was, we would be able to produce the Gödel sentence of that machine and would be able to intuit the truth of the sentence which would violate Gödel's theorem. The literature on this is long, but the consensus among most logicians is that the argument does not go through in the way that Lucas and Penrose envision it to.
So, assuming that the Lucas-Penrose argument is false and that our minds are Turing machines, your question is a very clear followup. It is important to note that not only are the traditional Gödel sentences (using his Bew( ) function about provability) apparently an issue for our mind, but also related undecidability results about recursively enumerable sets such as the halting problem, of course then also the Entscheidungsproblem, and as a result of the work on Hilbert's 10th problem and the MRDP theorem about diophantine sets as well. The work in recursion theory that showed all of these problems are related implies that there are many different sentences that would be undecidable to humans if human minds are indeed Turing machines.
In regards to whether or not someone would need a stronger, meta-meta logic that is inaccessible to human beings, the answer is no if you mean a different logic in respect to how first-order and second-order logic are different, but yes if you mean how Peano arithmetic and ZFC set theory are different. By which I mean, some being other than humans could still use first-order logic, they would just need to use a stronger system of axioms than we are able to see the truth of.
Imagine that ZFC is the ultimate limit of human comprehension. Then, we would not be able to prove the consistency or see the consistency of ZFC because of Gödel's second incompleteness theorem (of course this is again still assuming that we fall under the category of formal systems that his theorem applies to) unless ZFC and hence ourselves are inconsistent. However, there are some formal systems that do prove ZFC is consistent, such as Morse-Kelley set theory or systems of ZFC + a large cardinal axiom. This is because these systems are stronger than ZFC, they make stronger assumptions in their axioms. So if there was some being whose limit of comprehension was MK set theory for example, then they would be able to see the consistency of a human being assuming we are consistent. MK set theory is first order axiomizable, so there isn't a need to go beyond first order logic, the being just needs stronger axioms.
Consider this example as well. In a fascinating paper titled A Relatively Small Turing Machine Whose Behavior Is Independent
of Set Theory, Adam Yedidia and Scott Aaronson prove, among other things, that the 7,918th busy beaver number is independent of ZFC by creating a 7,918 state (later improved to 1919 states) Turing machine that enumerates all of the theorems of ZFC and halts if and only if there is a contradiction. If this Turing machine never halts, then ZFC never proves a contradiction and is therefore consistent, however if ZFC were to prove that this Turing machine never halts, then it would prove it's own consistency and therefore violate Gödel's second incompleteness theorem. This means that if ZFC is consistent, it will never prove that the Turing machine doesn't halt.
However, MK set theory can prove that the Turing machine never halts because "ZFC is consistent" is a theorem of MK. So if MK set theory is consistent and sound with regard to pi 0 1 sentences (in this case meaning, when MK says that a Turing machine doesn't halt, that Turing machine does not actually halt), then we know that ZFC is consistent as well.
So to answer your question, imagine that we have a set of axioms that are the axioms of the formal system of the human mind (again, under the assumption that the mind works that way). Then, we could construct a Turing machine that will enumerate all of the theorems of those axioms and halt if and only if there is a contradiction, just like in the above example with ZFC. If our mind's axioms are consistent, then that Turing machine will never halt. However, we can never prove that the Turing machine will never halt, because that would violate Gödel's second incompleteness theorem which we are subject to given the stipulations about our mind. But just like with ZFC again, any system that could prove our axioms consistent would be able to prove that the Turing machine does halt, we just can't do it ourselves. In fact, as Hilary Putnam (the P in MRDP) has argued (I don't remember the exact paper but if someone knows please comment it and if I find it I'll update my answer), if we were a Turing machine we wouldn't even be able to tell which Turing machine we were! So would we need a meta-meta-logic to study ourselves? No, we wouldn't even be able to study ourselves or to recognize which machine represented ourselves. Any being that has stronger mathematical capabilities would be able to study us, but they wouldn't need anything other than first order logic and a strong axiomatic system that we could never see the truth of.
Let me say two further things about the nature of what our minds being Turing machines really is. First, Martin Davis (the D in MRDP) has said in his discussion of the Lucas-Penrose argument that there is a very, very important detail that is being looked over. Gödel's theorems, the halting problem, the MRDP theorem, etc. only apply to us if we are consistent formal theories. Remember that Gödel's theorem only applies to recursively axiomizable, omega-consistent (a halfway point between consistency and soundness) formal theories that have enough power to interpret Peano arithmetic (Rosser later simplified the result to only need consistency, be recursively axiomizable, and to interpret Robinson arithmetic). It is perfectly possible for our minds to not be consistent, people have certainly believed inconsistent things before. As Davis points out in his essay on this topic Is Mathematical Insight Algorithmic?:
Note that the [Gödel] sentence is proved by an algorithm. If insight is involved, it must be in convincing oneself that the given axioms are indeed consistent, since otherwise we will have no reason to believe that the Gödel sentence is true. But here things are quite murky: great logicians (Frege, Curry, Church, Quine, Rosser) have managed to propose quite serious systems of logic which later have turned out to be inconsistent. "Insight" didn't help.
Additionally, in a personal correspondence I had with him on the history of this topic, he stated:
My own position (that of Turing in the 1950s) is to note that "we could prove a theorem it does not prove," only if we know that the machine's output is consistent.
So it is not at all plainly evident that we should consider ourselves consistent formal theories, even if we are Turing machines. It could be the case that our mind works algorithmically, obeying the Church-Turing thesis on a physical level, but that our axioms are inconsistent and therefore we sometimes will produce false statements even if we follow an algorithm. This is a subtle point and deserves a lot of attention when discussing these matters.
The final point, which really brings it all the way back to the disclaimer at the beginning, is that it isn't at all clear what "our mind is a Turing machine" means. Personally, I am a subscriber to the computational theory of mind but I also recognize that there is philosophical, as well as neuroscientific, work left to be done in regards to fully fleshing out what this position says and how we should understand it. In a paper, Incompleteness, Mechanism, and Optimism which surveys the Lucas-Penrose argument, Stewart Shapiro opens the section on the nature of the computational theory of mind (mechanist's thesis) thusly:
One problem is that the exact content of the mechanistic thesis is usually left unspecified. To belabor the obvious, the relevance of the incompleteness theorems to mechanism depends on what the mechanist
claims. The raw thesis that the human mind is, or can be modeled as, a
digital computer or Turing machine, is too vague to apply anything as sharp and delicate as the Gödel theorem and the Turing-Feferman extensions. My conclusion (perhaps slightly exaggerated) is that there is no plausible mechanist thesis on offer that is sufficiently precise to be undermined by the incompleteness theorems.
The results of what our mind can and cannot prove if we are formally equivalent to a Turing machine and a set of axioms is very interesting, but it requires idealizing the human mind to such an extreme degree that it becomes very unclear what exactly we mean. I haven't mentioned it yet but this is why none of the results gone over above have any impact on materialism. If the human mind is a Turing machine, all it takes is some being smarter than us whose internal mathematical axioms are stronger than ours and can prove the consistency of ours. There doesn't need to be a supernatural being involved, just some being that is smarter than us. This points out an even harder question for mechanists, who do we choose to idealized? There are obviously some people who have less competence in mathematics than others. Do we take the smartest human and idealize their mind to an axiomatic system? Do we take the average of all humans who are alive at this point? What if humans continue to evolve, how would changes in our brain change this topic? These are some of the problems in idealization that Shapiro points out.
There is ample literature on all of these discussions and you should follow the bibliographies in the linked posts if you are interested in more. I would highly recommend The Godel Theorem and Human Nature by Hilary Putnam (this might even be where that quote I couldn't place earlier is from).