I asking where is it "claimed by the Romanists that Paul went to Peter in Galatians 2 to "check" that he was teaching correct doctrine."
I believe that you have made such a claim.
It is claimed by Rome (the RCC) that Peter (and his successors), as "pope" is the sole "leader and representative" of Jesus Christ on earth. Paul consistently, throughout his epistles, proves that claim to be FALSE. Paul claims equal but different authority. Paul is the founder of the body of Christ based on the authority given him by the RISEN and ASCENDED Lord Jesus Christ.
Peter was ONE of TWELVE apostles that have authority over ISRAEL. Israel is FALLEN.