In the context of logical argumentation, they move from being a Logician or a Philosopher to a Rhetorician, a Sophist, or just an Antagonist depending on their goal being dismissive.
Regardless, if you have caught someone in a contradiction in a logical debate and you call it out, the argument is over - if the situation requires them to look or read something because you tell them they have contradicted themselves...and all they need to do is look; refusal is tantamount to them looking and recognizing the contradiction. They are a Sophist, a Brigand, or the Loser of a Debate by Default.
If they dismiss the contradiction without even addressing it, yet aggressively push the argument forward, they are the Loser by Default, but they are also an Antagonist or an Oppugner (this is rooted the same as a Pugilist; therefore, although not widely used, I think it is an apt term).
But if the debate doesn't end once they've contradicted themselves, you essentially have the green light to counter with anything, including an ad hominem attack, because, logically, anything follows from reductio ad absurdum. It might be out of poor taste to do this, but that depends on context.
For example, if you're debating someone in prison who dismisses the contradiction and begins aggressively arguing forward, you can't just walk away because you won. That makes you a punk; and nobody wants to be a punk in prison - so in this case, to prove a point, you are released from the rules of logic and should hammer away.