I agree with default locale that the dialogue is probably the Gorgias and the interlocutor is Callicles.
Socrates presents himself throughout the
Gorgias as the great defender of justice, the unwavering advocate
of the extreme pro-justice view. Especially in the final part of the
dialogue, once Callicles has refused to take part seriously in the
conversation and enters only intermittently, this view takes center stage and dominates the several long speeches Socrates gives
(see, e.g., 507a5-509c4, 511b7-513c3, 522c7ff.) (Devin Stauffer, 'Socrates and Callicles: A Reading of Plato's "Gorgias"', The Review of Politics, Vol. 64, No. 4 (Autumn, 2002), pp. 627-657: 651.)
But in a way we can also say that Socrates does not participate in the dialogue fully, because he does not consider Callicles to be capable of cogent, connected philosophical argument :
Throughout their conversation,
Socrates has to make great efforts to ...
bring to the surface what Callicles really believes. Callicles' resistance to Socrates' attempts to do this is not merely a matter of shame
at making concessions to Socrates, or a way of protecting a facade he
has consciously put in place, but rather it reflects an unwillingness
to acknowledge, even to himself, concerns that he has dealt with by
trying to sweep them under the rug. Since Callicles is so unwilling to
face up to what he really believes, we can thus understand why Socrates would think that it would be futile to try to engage him in the
kind of examination that is the true heart of a Socratic education.
Callicles is not capable of such an examination because he is dishonest with himself about what he really believes. (Stauffer: 648.)
If this reading is correct, Callicles is not showing or attempting to show that there is anything deeply wrong with the Socratic method. He is not outright critical of the method from the start, as is Thrasymachus in Republic I. He is rather avoiding the critical self-examination to which that method is driving him. He does not really critique the Socratic but rather impeaches his own intellectual seriousness, if Stauffer's line is right. In this sense, different probably from that of your question, Callicles shows bad faith.
In sum, I think you have Callicles in mind but I am not so sure that he exactly challenges the Socratic method even indirectly. But you must weigh what I've said against your own impressions and compare it with other answers. Welcome to PSE.