Saturninus and Bassanius are already addressing the assembled senators and explaining why each thinks he should be the next emperor. When they agree to give up their claims in favor of Titus Andronicus becoming emperor, it sounds to me like they leave to go to the capitol and let everyone know of their decision, but the text doesn't say anything about them becoming senators. I definitely don't believe that would have been Saturninus' decision as he had every intention of coming back and asserting his right to be emperor:
"Romans, do me right:
Patricians, draw your swords: and sheathe them not
Till Saturninus be Rome's emperor.
Andronicus, would thou wert shipp'd to hell,
Rather than rob me of the people's hearts!" (I, i)
Both Bassanius and Saturninus were sons of the late Roman emperor, so they both had a claim to the throne. Their ultimate goal would have been to achieve that throne, not to become part of the senatorial class.