no, they would just need to each have at least one member reach enough to be on the album, which until the rise of the Equites, I would be surprised to learn they didnt'.
"Senatus PopulusQue Romanum" "The Senate And the Roman People" as in the two are distinct and separate.
"The Senate of the People of Rome" would be "Senatus Populi Roma" and thus SPR, not SPQR.
The Senate, I stand by that, was meant to represent the original gentes who evolved into the patricians. True, social pressure forced it to be more inclusive than that, but it was never the assembly of the roman people, that's the comices' job (and the plebeian counterpart to the Senat was the plebeian council).