First you say there is no evidence, then you say there IS evidence.
Whether there is information in tradition about Bartholomew being persecuted, that doesn't have anything to do with there being historical evidence that Paul was persecuted for following Jesus. Paul’s Radical Conversion To Christianity | Reasons for Jesus
Thirdly, Paul lived his post-conversion life with a willingness to suffer and be persecuted for the message of Jesus. This is quite the dramatic change considering that his pre-conversion life had him persecuting and killing Christians. Not only did Paul suffer abuse, persecution, heartache, and isolation but he was also later martyred. On several occasions Paul was incarcerated, while other times he endured beatings (2 Cor. 11:24-27).
The Jews with whom Paul spoke tried to kill him (Acts 9:29), he was persecuted (Acts 13:50, 1 Corinthians 4:12, 2 Corinthians 4:9, 2 Timothy 3:11 & Phil 1:12-30), he was stoned and dragged out of the city (Acts 14:9), beaten with rods (Acts 16:22), endured trial (Acts 18:12), verbally abused by crowds (Acts 21:36 & 22:22), and incarcerated (2 Timothy 2:9).
Paul was clearly willing to suffer for his faith as is multiply & independently attested. From our early New Testament writings Paul’s suffering is attested by his authentic letters (1 Corinthians etc.), disputed letters (2 Timothy etc.) and by Acts.
That is at least three independent traditions attesting to his persecution and willingness to suffer for the risen Jesus. We should clarify one detail, however. To have two or three independent sources on Paul’s suffering depends on whether or not one holds that 2 Timothy is an authentic or disputed epistle of Paul’s. According to roughly 80% of scholars 2 Timothy was not authored by Paul (3).
We will accept this consensus, and therefore we have three independent sources from within our New Testament attesting to the persecution and suffering of Paul. According to theologian Chris Price:
“Most striking perhaps is that fact that Jesus appeared to Paul. Paul hated Christians and was hell-bent on destroying the church. What transformed him from a persecutor of Christians to a pastor, who was willing to endure extraordinary hardship to proclaim the Gospel? Paul claimed it was the resurrection. This also indicates that Jesus didn’t just appear to friends or followers who might have been predisposed to think high and exalted things about him. Christ appeared to skeptics (James) and unbelievers (Paul) and they were convinced based on the reality of the resurrection” (4).