The barber is our no-nonsense character brought up just for a sec at the start of the text as a foil to the mute, then gets more airtime when shaving Frank. He's easily the most reasonable character in the text, which might be why his devil's bargain doesn't get the same kind of play-by-play everyone else's does:
...the cosmopolitan rose, and, for added refreshment, washed his face and hands; and having generally readjusted himself, began, at last, addressing the barber in a manner different, singularly so, from his previous one. Hard to say exactly what the manner was, any more than to hint it was a sort of magical; in a benign way, not wholly unlike the manner, fabled or otherwise, of certain creatures in nature, which have the power of persuasive fascination—the power of holding another creature by the button of the eye, as it were, despite the serious disinclination, and, indeed, earnest protest, of the victim. With this manner the conclusion of the matter was not out of keeping; for, in the end, all argument and expostulation proved vain, the barber being irresistibly persuaded to agree to try, for the remainder of the present trip, the experiment of trusting men, as both phrased it. (43, 24)
When the other characters lose money, the conversation is painstakingly rendered for us. Now all of a sudden the text deliberately obscures how Frank got the barber to agree to taking down the "No Trust" sign we see him put up early on. What we do get is a sense that there's something unnatural afoot. The narrator specifically uses terms like "fascination" and "magical" to describe the power Frank has over the barber. Frank's like a supernatural creature, and only that way can he convince the barber to go against his own nature.