Word for someone who remains calm (and taunts) in arguments

You can use:

self-controlled (adjective usually approving)

having strong control over your emotions and actions.

”He was self controlled even when the English teacher yelled at him.”

(Cambridge Dictionary)

I think composed can be used in this context, as it is a synonym for the word calm itself.

Equanimous among countless other synonyms of the word would fit the bill.

If you're equanimous, you're serene and calm. People who are equanimous seem enviably composed and balanced.

[Vocabulary.com]

It's sort of depends on the context.

For example, my father was famous for his stoicism when my mother argued with him, which of course made her angrier and ultimately violent.

But over time the blood would drain from my father's face and his expression hardened in defiance, but we all knew he was about to lose his $hit."

But my sister would remain calm in an argument, with a sarcastic expression, is if she thought I was foolish."