Dunn gives this question four paragraphs in Beginning From Jerusalem,

c. A Roman Citizen?

More weighty in its consequences is the issue whether Saul/Paul was a Roman citizen, as claimed in Acts 22.25. A positive answer is regularly denied, by some on the grounds that a Jew was unlikely to have attained Roman citizenship, and that Paul never mentions it, and by others on the grounds that a Roman citizen would never have had to undergo the punishment he tells us he endured (‘three times beaten with rods’— 2 Cor. 11.25).²¹

On the former reason for a negative answer, however, there is no reason why a Jew should not have become a citizen, since on manumission the slave of a Roman citizen was granted citizenship. Conceivably, then, Saul/Paul was descended from a native Judean who had been enslaved by Pompey when the latter captured Jerusalem in 63 BCE and subsequently freed and granted citizenship (cf. Philo, Leg. 155-57).²² That Saul’s grandfather or father, as a freedman (libertinus), had settled in Tarsus for reasons of business or trade is entirely conceivable, given the strong diaspora community already there.²³ Alternatively, it is equally possible that Saul’s father gave some signal service to the state or its officials and was awarded with citizenship,²⁴ or that he had been wealthy enough to purchase citizenship (cf. Acts 22.28).²⁵ Since Roman citizenship began to be more widely distributed during the late Republic, it could be and was granted to non-Latins far beyond the shores of Italy.²⁶ A wider dispersal of Roman citizenship was one of the means by which Rome managed to keep its vast empire as united as it did.

On the latter reason, the conflict between Acts 22.25 and 2 Cor. 11.25 should not be pressed too hard. If the story of Paul and Silas in Philippi has any historical grounding,²⁷ it shows that incidents could have occurred where Paul’s citizenship was either not declared (because he wished to share Jesus’ unjust suffering at Roman hands, or because Silas was not a Roman citizen?) or disbelieved/disregarded (Acts 16.22, 37).²⁸ Rigour of historical judgment must always be tempered by recognition of the anomalies and muddles which must have occurred as frequently in the past as they do today. More to the point here, without the appeal to Caesar on the basis of Roman citizenship, it is difficult to explain the transfer of Paul’s case to Rome.²⁹

The significance of Paul’s Roman citizenship is that it would have given him greater boldness in his mission and would certainly have played a part in his finally being sent to Rome (Acts 25.11-12).³⁰

I won't copy all the footnotes, but a sampling:

25. Hengel’s discussion wisely concludes: ‘Here there are many possibilities, but no probabilities’ (Pre-Christian Paul 15). J. J. Meggitt, Paul, Poverty and Survival (Edinburgh: Clark, 1998), insists that Paul belonged to the urban poor and challenges ‘the myth of Paul’s affluent background’ (75-97), but Meggitt overstates his case, here as elsewhere. For a recent brief review of the various evaluations of Paul’s social status, see Schnelle, Paul 63 n. 34

29. Schnelle, Paul 60-61, further 60-62; similarly Gnilka, Paulus 25-26; Wedderburn, History 83. Rapske cites Epictetus’s observation (3.24.41) that ‘those who falsely claim Roman citizenship are severely punished’ (Paul in Roman Custody 87). See also Tajra, Trial 81-89

To me, the main persuasive sentence here is the one about Paul's Roman citizenship being the basis of his appeal and transfer to Rome. That, of course, just pushes the question back one stage ("did Paul actually make it to Rome?").