The Split between Paul and Barnabas

I have been wondering recently whether Barnabas might help us with some of the vexed problems of sorting out Paul's chronology. Paul and Luke are agreed that there was a major split between Paul and Barnabas, and they are agreed on the timing, not long after the Jerusalem Council (Acts 15 // Gal. 2.1-10), though they disagree on the content of the dispute between them:

Gal. 2. 11. But when Cephas came to Antioch, I opposed him to his face, because he stood condemned. 12. For prior to the coming of certain men from James, he used to eat with the Gentiles; but when they came, he began to withdraw and hold himself aloof, fearing the party of the circumcision. 13. The rest of the Judeans joined him in hypocrisy, with the result that even Barnabas was carried away by their hypocrisy.

Acts 15. 36 Some time later Paul said to Barnabas, "Let us go back and visit the brothers in all the towns where we preached the word of the Lord and see how they are doing." 37 Barnabas wanted to take John, also called Mark, with them, 38 but Paul did not think it wise to take him, because he had deserted them in Pamphylia and had not continued with them in the work. 39 They had such a sharp disagreement that they parted company. Barnabas took Mark and sailed for Cyprus, 40 but Paul chose Silas and left, commended by the brothers to the grace of the Lord.

He never appears again in Acts, nor does he appear again in Paul's correspondence except in 1 Corinthians 9, one tantalising, brief reference:

1 Cor. 9.6: “Or do only Barnabas and I not have a right to refrain from working?”

It looks like, when writing 1 Corinthians, Paul was still friendly with Barnabas. But if that is the case, we have only a narrow window for the writing of 1 Corinthians. The letter is clearly written after the Jerusalem Council since the collection is now underway (see 1 Cor. 16), but if it is also written while Paul is still friendly with Barnabas, it is written in that narrow window between the events reported in Gal. 2.1-10 (Jerusalem Council) and Gal. 2.11-20 (Dispute at Antioch). Is that a large enough window?