You might be over-thinking this one.
Paul uses the word my to emphasize the difference between his calling to preach the gospel to the uncircumcised and (say) the apostle Peter's calling to preach the gospel to the circumcised.
Paul was destined from birth to be the first apostle to be called by God to preach the gospel to the Gentiles. That was the ministry God entrusted to him. The content of the gospel, whether to Jews or Gentiles, was the same for both audiences. Generally speaking, however, the Jews who became followers of Jesus Christ did not imagine non-Jews as even being worthy of the gospel, as Peter demonstrated so well in Acts Chapter 10.
So entrenched was Peter in believing Gentiles were "unclean," God had to give him the same vision three times! He finally got the message, and he received the attitude adjustment he so desperately needed. He went from saying "Not so, Lord," to "I should not call anyone impure or unclean."
In short, then, "Paul's gospel" did not belong to him; it was part and parcel of his ministry to his primary audience, the Gentiles. Furthermore, until the believing Jews, including Peter, caught up with God, so to speak, and were on the same page as God regarding the Gentiles, Paul would find himself defending his ministry and its message--the gospel--that in Christ
there is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus. If you belong to Christ, then you are Abraham’s seed, and heirs according to the promise (Galatians 3:28-29 NIV).
In conclusion, "Paul's gospel" was not the only extant gospel to be found in the province of Galatia. There were other gospels making the rounds. Paul needed to make very clear to the Galatian believers that the gospel he was preaching--his gospel--was the genuine article, prompting him to say,
I am astonished that you are so quickly deserting the one who called you to live in the grace of Christ and are turning to a different gospel—which is really no gospel at all. Evidently some people are throwing you into confusion and are trying to pervert the gospel of Christ. But even if we or an angel from heaven should preach a gospel other than the one we preached to you, let them be under God’s curse! As we have already said, so now I say again: If anybody is preaching to you a gospel other than what you accepted, let them be under God’s curse! (1:6-9 NIV, my emphasis).