The Genovese leadership during Carlo's reign, and especially during Castellano's, was always friendly with the Gambinos, so I think war is a rather unrealistic scenario for that time period.
Your scenario was from before Castellano was killed. Chin was friends with Big Paul. But, he despised Gotti. That, combined with the self serving rule of never whacking a mob boss, should have sealed Gotti's fate. The Gambinos were weaker than the Genovese/Lucchese combined. Once Gotti was gone, the beef would have been settled.
DeCiccio and Lino were clipped by Chin/Casso-Amuso. And Angelo Ruggiero would have been murdered by Casso, since he was the one that sent Jimmy Hydell after Gaspipe. And they were after Sammy the Bull, but he was too evasive. So, in a way, there was a war going on.
