But if the Corleones agreed to completely support Sollozzo, why should he still conspire with Barzini? That wouldn't be in his interest anymore.

Besides, the Corleones were the most powerful family at that time and the Tattaglias were generally held in disregard. As Sollozzo was a businessman why should he not be loyal to the Corleones if they gave him everything he wanted. That would have been his best bet, logically speaking.