If Paul was writing under the inspiration of the Holy Spirit (and he was), we can be assured that the incident took place and was accurately recorded.
I'm not sure that the Matthew 18 reconciliation process is applicable to the situation of false public teaching. Even if it is, isn't quite possible that Paul had already discussed this with Peter privately and what is recorded in Galatians is step 2 of the process: take others and confront?
If Paul was writing under the inspiration of the Holy Spirit (and he was), we can be assured that the incident took place and was accurately recorded.
12 For before that certain came from James, he did eat with the Gentiles: but when they were come, he withdrew and separated himself, fearing them which were of the circumcision.
Why the remark about James? it is recorded in the same chapter of acts that James agreed with Peter that the gentiles was given the same grace as the Jews.
It was James who had a letter written to that affect to send with chosen men to return with Barnabas and Paul to Antioch.
I certainly hope Paul was not being inspired by God to slander the name of any of Christs hand picked apostles if that is true we are in trouble.
Paul had great authority before he became a Christian, it appears to me that he could not let go of it.