Heyting arithmetic (a theory built entirely on intuitionistic first-order logic, which lacks the law of excluded middle, interdefinability of quantifiers, and De Morgan duality) can prove cut-elimination for classical propositional logic, and classical first-order logic.
This isn't circular!
In a proof of cut-elimination for classical logic that's formalized in Heyting arithmetic, one starts with a mathematical axiom system formulated in an altogether different logic! This answers your object level question in the negative: it is not true that you can only prove cut-elimination for the sequent calculus by using classical logic in the metatheory.
Heyting arithmetic is just one possible example, of course. Logicians have developed a wide variety of other foundational systems that either use a nonclassical logical substrate, or don't require a separate logical substrate at all (type theories with propositions as types). These foundational systems range from widely used (Martin-Löf's type theory, the Calculus of Constructions) to niche and underexplored (extensions of Light Affine Set Theory) to completely obscure. A great many of them can prove cut-elimination for the classical sequent calculus despite not being classical themselves, and they can provide usable meta-theories should you need one.
Unlike the results of set-theoretic model theory, which fail big time the moment you drop even just the Axiom of Choice from ZFC, proof-theoretic results such as cut-elimination tend to be quite resilient: we can usually justify them through many different meta theories.
Is this circular?
You might find all this quite unsatisfying anyway. After all, in our example, Heyting arithmetic seems stronger than first-order logic in a certain hard-to-quantify sense. For example, in the sense that the former proves the consistency of the latter, while the latter does not prove the consistency of the former.
And if we choose Heyting arithmetic as our metatheory, don't we make a precommitment to intuitionistic logic, which also requires its own cut-elimination and consistency result?
Generally, when we decide on a foundational theory which we use as basis for our deductions, we'll have some justification for choosing it. This justification could have many sources: a philosophical argument, divine revelation, history and tradition, parsimony, inertia, "we empirically checked and this one looked alright", etc. And the justification will sometimes compel us to think that some particular logic-like structure or deductive apparatus is good (but this need not be an outright assumption of consistency; plenty of people decide to work in ZFC even though they admit that they don't know whether it's consistent or not).
If you then attempt to use the chosen system to prove the consistency of its own deductive/logical substrate, and you succeed, you'll still have more justification for it than before: e.g. you'll understand better where the load-bearing principles are, and how other starting points could also justify your theory (or not). More importantly, it's not a given that you'll even be able to do it! For example, you could just as well formulate the Peano axioms using the second-order sequent calculus as the logical substrate: and then you'd observe that you can't prove the consistency of your logical substrate, no matter how hard you try, and proving cut-elimination will remain forever out of reach.
In any case, what you can't get out of these arguments is justification for your original commitment ex nihilio. The good news is that nobody has ever demonstrated any way of getting anything ex nihilo anyway. The bad news is that no matter how many proofs you have, it can still happen in principle that classical logic happens to be inconsistent, and then its proofs of consistency are all worthless. But then almost all known proofs are worthless, and in any case that's not an issue of circularity. Which brings me to my last point.
What is circular?
Since this is a proof theory question, let's discuss logical terminology a bit. If you work in some classical first-order meta-theory, and you have a proof inside that theory which
- starts from axioms of your first-order theory, and
- uses only deductive inferences that are valid in classical first-order logic,
then that proof is not circular. This is true irrespective of the subject matter your proof is about, or the particular conclusion that it proves.
For example, if your meta-theory is ZFC, and you use it to give a valid proof of Zorn's lemma, there's no sense in which that proof is circular. Similarly, if you write a ZFC proof that first proves Zorn's lemma from the ZFC axioms, then in turn uses the freshly proven Zorn's lemma to deduce the Axiom of Choice, that proof is still not circular. It's superfluous, perhaps (you could have concluded the same thing before step 1), but not circular in the technical sense. Indeed, as far as mathematics is concerned, there is absolutely nothing wrong with such a proof.
The term "circular proof" instead refers to something that looks superficially or locally like a proof, but cannot be turned into an actual proof, because it justifies one of its steps ultimately in terms of itself. In sequent calculus, this would correspond to a never-ending proof tree (e.g. imagine cutting against $B$ while trying to prove $\vdash B$, yielding the two goals $B\vdash B$ and $\vdash B$ again; repeat ad libitum). Circular proofs are not formally valid proofs; unlike the proof of consistency of classical first-order logic from the Peano axioms, which is definitely a formally valid proof. Depending on your philosophical views, the proof might not be reassuring, but it's a formally valid deduction nonetheless.
In the technical sense, a proof that ends with proper axioms cannot be circular: the axioms are axioms because they don't have to be further justified. It might happen that you chose your axioms because you believed them to be consistent: or chose the logical substrate of classical logic because you believed that to be consistent. The proof is a sequence of symbols, it can't tell and doesn't care how you chose your axioms or your logical substrate, and its circularity cannot depend on what justifications you in particular had for the axioms you chose.
As I explain in detail elsewhere, formal axiomatic proof does constitute the gold standard of mathematical argumentation: derivations in which every step is justified by the rules of inference of a formal logic and by select foundational axioms of mathematics. But mathematicians don't write formal proofs, except perhaps when interacting with proof assistant software and occasionally in a university course dedicated to Proof Theory. The majority can't even state the foundational axioms. Instead, proofs in academic mathematics, in journals, textbooks, are rigorous informal proofs, communicated in free-form natural languages such as English. They may freely skip over tedious, repetitive or routine steps, and need not go back to axioms at all: they refer to well-known results and informal proofs of other results all the time!
The ultimate legitimacy of these informal proofs hinges on their ability to serve as mnemonics for constructing a formal proof: sure, it does not have all the details, but if a persistent interlocutor were to question the missing details, it would be possible to fill them in.
"Looping back" is a legitimacy problem only in as much as it prevents you from constructing the formal argument in principle. If you choose to prove the irrationality of the cube root of 3 by reducing it to Fermat's last theorem, and cite Wiles' proof as justification for Fermat's last theorem, and in turn Wiles' argument relies on establishing the irrationality of the cube root of 3 somewhere, that's not inherently wrong, even though it's neither parsimonious nor elegant. It'd make your proof illegitimate only if that particular line in Wiles' proof could not be further justified, or could only be justified by infinite regress and never actually getting around to producing that formal proof.
Note that the standards are different when we do mathematics and when we examine freshmen. In the latter case, we're not concerned about whether we could fill in the details in principle: we're concerned about whether the student learned the material and can fill in appropriate levels of detail autonomously. This is why we wouldn't accept the FLT proof of irrationality of the cube root of 3 from a student on an exam: a student who gives that answer hasn't demonstrated the ability to fill in the required details: even if you assume that they could miraculously fill in all the reasonable details in the proof that come before Wiles first invokes the irrationality of the cube root of 3, their answer certainly gives no indication that they'd succeed in filling in the details at that particular point. The way to point it out is by saying that "this proof is circular".
Edit: a followup question
This follow-up question arose in the comments: Is it ever useful or 'interesting' to prove things which you already presupposed when choosing the metatheory?
Yes, such proofs have plenty of important applications.
First of all, the fact that you can separate your foundational system $S$ into a simple logical substrate $S_c$ whose consistency you can prove inside $S$, and a different, axiomatic part $S_u$ whose consistency you cannot prove and have to take on faith, is already a nontrivial property which not all possible foundational systems obey. For instance, in a version of Peano arithmetic built upon impredicative second-order logic rather than classical first-order logic, attempting to verify the consistency of the natural logical substrate (i.e. impredicative second-order logic without axioms) would prove futile.
However, when such separations are possible, the specific ways you can divide $S$ into a segment that admits a consistency proofs, $S_c$, and another that does not, $S_u$, reveal much about the foundational system in question!
For example, imagine that you chose ordinary Peano arithmetic, formulated in classical first-order logic, as your meta-theory, based on some justification. Now, it's a celebrated result that Peano arithmetic can in fact prove the consistency of any fixed finite subsystem of itself. So you can always split Peano arithmetic $S$ into the system "first-order logic plus a finite, possibly large, set of induction axioms" $S_c$ whose consistency you can actually prove inside $S$, and another larger set of induction axioms whose consistency with $S_c$ you cannot prove.
Is such a proof good for anything? After all, your justification for choosing Peano arithmetic as your foundational theory presumably included the belief that it's consistent, and by extension the belief that all its finite subsystems are consistent as well. Nonetheless, this result offers fresh insight: since Peano arithmetic proves the consistency of all of its finite subsystems, but (you have the belief that) it does not prove its own consistency, you should now believe that Peano arithmetic is not a finite subsystem of itself, i.e. that it is not finitely axiomatizable. So the proof teaches you a highly nontrivial property of Peano arithmetic, and one that we absolutely did not know when we first formulated it (note that e.g. NBG set theory, a much stronger theory often used as a foundation, is finitely axiomatizable).
All in all, these proofs serve a useful purpose, and enable the extraction of substantial insights. It's just that these insights will not include absolute certainty regarding the consistency of the chosen axioms.
