The translation you quoted came from Weymouth New Testament, the only one using the word "sophistry" to translate the phrase "τίς ὑμᾶς ἐβάσκανεν ⧼τῇ ἀληθείᾳ μὴ πείθεσθαι⧽" (who hath bewitched you, that ye should not obey the truth) in Gal 3:1. It is insufficient evidence to say that Paul or NT believers had issue with sophistry, with the Sophists or with the philosophy that Sophists used.
I agree with your modern definition of sophistry and with the characterization of the sophists of ancient Rome. More fundamentally, the Sophists did not believe that there are objective moral truths, akin to today's moral relativists. But even worse, Sophists claim that justice is "nothing but the interest of the stronger" (see the article Is Justice a Matter of Truth or Power), again an idea that is distressingly still with us today. No wonder then, that this idea justified them to defend either side (the "guilty" as well as the "not-guilty") in the courtroom with powerful rhetoric unconstrainted by truth (i.e. with sophistry).
I don't think believers in Paul's era battled them; we only see Epicurean and Stoic philosophers in the NT. The most powerful philosophy battled by the early church fathers was Neo-Platonism, prompting early apologist Tertullian to exclaim his famous quote "What Has Jerusalem to do with Athens" although he didn't mind using philosophical terminologies to defend Christianity.