Not if Duce sucessors hadn’t switched sides
Good Analogy. Subtly is not a roman strength.I get how this could work in principle. But wouldn't it be highly likely that whatever the Rome regime *says* its deposition and arrest of Mussolini can only mean its trying to get out of the war pronto. An analogy that comes to mind is that in Yugoslavia one King joined the Axis. He was overthrown by a nationalist coup. Although the new King and government were careful in their statements, not saying they quit the Axis, Hitler figured "they're just waiting to screw me over, so it's time to squash them".
Much depends on what the Italian govt. actually was going to do, and or what they said they were going to do and how well they concealed their true intentions. If they told the Germans "This man is an incompetent ass, we will run the war more efficiently without him" and actively took steps in that direction, they may have been able to continue in the Axis. Or, had they decided to switch sides, they had done the same but made more effort at allaying German suspicions. Their failure to do either very well led to the German occupation.
German troops were already being strategically deployed throughout Italy (not just in the south) even prior to Benny's arrest