I looked the word used in Philippians 3:2 to make sure and it is in fact literally the word for dogs, as in four-footed creatures with wagging tails. I will have to say that to me, as to you, Paul’s use of this word for his opponents is pretty strong language. But then, Paul is dealing with a very strong and very bad situation with people who are trying to pervert the gospel by trying to bring in law-keeping and works salvation. They are denying the grace of God and denying the gospel of Jesus Christ. No wonder Paul uses strong words here.
Nevertheless, we ought to ask ourselves exactly how strong the language was for Jews in the first century. It may not have been exactly what it is for us. For example, in Acts 20, Paul describes those who are trying to destroy the church “wolves.” For Americans, living in the US in 2022 (forgive me if I am falsely assuming you are American), to be called a wolf is not really that offensive, but to be called a dog is. In India calling a woman a cow is a compliment. Not so much so in the US. So… My point is that Paul was absolutely justified in using strong language for a drastic situation, but he was not justified in using offensive and insulting language. The question is whether calling someone a dog in Paul’s setting was more like calling that same person a wolf. Might Paul have stepped over the line? If he were speaking English in the United States in the year 2022, he might have been stepping over that line.
So, the question is this: Was calling someone a dog for Paul more like us calling someone a wolf or more like calling them a dog or pig today? Honestly, I am not quite sure, but I am not willing to charge Paul with using offensive language in this case.
The point is this, what is offensive in one place and time may not be offensive in another place and time. I am assuming that Paul did not use an inappropriately offensive term.
A side note. This is not the only time the word dog is used of humans in the New Testament. In Matthew 15:28 Jesus did not call the Canaanite woman a dog, but he applied it indirectly to the non-Jews. In this case, the Greek word is that for a small, domesticate dog, whereas the word in Philippians 3:2 is a stronger word, meaning a possibly wild dog. Jesus used a less offensive word of the Gentiles in Matthew 15 than Paul used for the heretical Judaizers in Philippians 3:2.