At the time of Caesar, senators were all Italian. In fact, one of Caesar's most controversial acts as dictator was expanding the senate and including men from Southern Gaul among its members. These men would have been culturally Roman and Roman citizens, but this was still a big deal at the time.

So no, there was no attempt at representation of the provinces within the senate. Neither Britannia or Dacia were part of the empire at the time of Caesar's civil wars, incidentally. Many Italian senators, however, had clients throughout the provinces, especially in places they had fought or governed. These clients were generally local elites. So, Cicero was asked by his clients in Sicily to represent them in an extortion case against the corrupt governor Verres in 70 BC. Pompey had many clients in Spain, including the very influential Lucius Cornelius Balbus (a citizen). Otherwise, local communities and lical elites throughout the empire had to appeal to their governor, who could change every year, and perhaps through him to the senate. Cicero's letters from Cilicia from 51-50 BC have a lot of info here.

Over time, the senate would become less Italo-centric. Particularly in the aftermath of Octavian's civil wars, a large part of the traditional senatorial elite was decimated. So, we start seeing prominent senators originally from the province, such as Lucius Cornelius Balbus the younger, son of Pompey's client and the last non-imperial celebrator of a triumph (in 19 BC) until Belisarius 500 years later. Eventually there would even be emperors from the provinces, beginning with Trajan. But even then, the Roman senate was never meant to be a representative body.