In the novel, Puzo described Barzini as the most "modern" Don, younger, colder, contemptuous of the "Moustache Petes." He also said that Barzini's ambition was to supplant Vito as the top Don, not necessarily to take over Vito's family. I infer that Barzini would want to keep killing to a minimum. So:
--Clemenza would have to go. He expected to be head of his own family, and would never settle to be subordinate to Tessio or Barzini. He'd be a mortal danger to both.
--Neri and Rocco were joined at the hip to Michael and Clemenza. They'd have to be killed at the same time.
--I think Barzini would want Tom to work for him. He'd need to continue the Corleone police/political contacts to protect his drugs business, and Tom would be the only remaining family member who could deliver them. Barzini might conclude that since Tom wasn't Italian and was a lawyer by training, he'd put pragmatism over loyalty. He'd probably "make him an offer he can't refuse." Tom would recognize that if he turned it down, he'd be dead. I think he'd take it--and work 24x7 to keep himself as the political go-between because once Barzini got the judges and the politicos wired to him, Tom would be superfluous--dead.
--I think Barzini would figure Fredo wasn't worth killing. His ally Moe Green had Fredo under control and could whack him at any time. Besides, Fredo had already shown signs of drifting loyalty. He was harmless, and might have been marginally useful.
--Unlike others, I think Barzini would have let Tessio live. Tessio would now become Barzini's protege, and it would be easier for Tess to run the Corleone enterprise under Barzini's thumb than for Barzini to do it himself. As Olivant advised, look to other Mob internal upheavals. Vito Genovese got Carlo Gambino to conspire against Albert Anastasia in return for becoming a Don; then Gambino became Genovese's protege and ally on the Commission until Genovese went to prison. Then Gambino got Joe Columbo to betray Joe Magliocco; Columbo become Don of the former Profaci family, and was Gambino's protege and ally on the Commission until he started up the Italian-American Civil Rights League and incurred Gambino's wrath. No large-scale massacres followed either the Maranzano coup or Castellano's murder, either. As Solozzo said, "Blood is a big expense."