The Romans did, in fact, have slurs or at least phrases that were demeaning of Carthaginians or punic culture in general. I can't seem to find a source with reference to slurs as we know them: one word that is used in place for the entire people. The phrases appear to be more focused on the stereotypical behavior of Carthaginians. For instance, we see multiple derogatory remarks about the Carthaginians in the works of Livy. One such term that was common in the Augustan period was "fides Punica" which literally means "Punic faith." However, this phrase actually referred to faithlessness, implying that the Carthaginians were faithless people. Additionally, in Virgil's Aeneid, the line "domum timet ambiguam Tyriosque bilinguis" appears when Dido fears allowing typical Carthaginian behavior to occur. Essentially, the phrase implies ambiguous/deliberately dishonest language being spoken by Carthaginians. This represents a Roman stereotype of Carthaginians as liars or "fork-tongued."

Unfortunately, as I stated previously, I was unable to find really any "names" given to the Carthaginians during the Punic Wars in the same way Americans/Brits called Germans "krauts" during the Second World War. However, these sources do imply that stereotypical views of Carthaginians were held by the Romans and, in my opinion, it's not unlikely for such slurs to have existed in the Roman/Latin vernacular. Hope this helps!

Note: First time doing this so any advice would be appreciated.

Source: "Fides Aeneia: The Transference of Punic Stereotypes in the Aeneid" by John H Starks, Jr. https://www.jstor.org/stable/pdf/3298369.pdf