The weakness of MAD is that using it is by definition a suicide, so as long as your enemy does not believe your actions threaten their existence the threshold to unleash the nuclear weapons is quite high. During the Cold War this showed in a large number of proxy wars in what was usually called the third world. In general the concept is to have a clear idea what are the triggers for the other side and then do something else. Bonus points if your idea is related to reality.
In practice, if you want the evil guys to lose you can model their leader after early Hitler. He started with reoccupying Rhineland which basically everyone agreed was justified even if potentially a dangerous sign. Then unification of Austria, which nobody could oppose too much as the Austrians had historically been considered Germans. Then Sudetenland which Hitler was able to get away with because the borders created after the war were still kind of artificial and because nobody else wanted war. Which as I mentioned is the weak point of MAD.
The reason this is a losing strategy is because it is a dead end. You get away with it once on some relatively minor issue and then everybody is seriously thinking that they'd might have to pull the trigger if "that crazy person" doesn't stop. And stopping will be very hard with your confidence sky high and everybody telling you what a great leader you are. And since nobody wants to trigger a war, they'll avoid anything provocative, or with other words are looking weak and clueless.
If you want the bad guys to win model the leader after Bismarck. Everybody knew Prussian militarism and Bismarck were dangerous and must be contained. It didn't do them any good. For example in your scenario typical leader would see themselves surrounded by enemies and act accordingly. Pseudo-Bismarck would realize that divisions such as allies and enemies are largely meaningless as they depend on the situation and the situation is influenced by your own actions. In the correct situation your steadfast enemy would end up neutral or even be on your side. Typical leader would act tough against enemies to avoid being taken advantage of and the enemies would close ranks against him. Pseudo-Bismarck would have no need to act tough and would freely engage with anyone willing to talk with him in order to keep the politics fluid and complex. Read exploitable. Typical leaders react to unexpected events by playing it safe, doing the expected, and avoiding mistakes. Pseudo-Bismarck expects the unexpected to happen, can predict what other leaders will do in response, and always has several goals he can push forward in the opening created by his potential opponents being committed on a strategy he can predict.
I could go on, but somehow I doubt you want the evil empire to win and need more detail on this alternative. Pseudo-Bismarck would not let the world go MAD, pseudo-Hitler would almost guarantee that world will go MAD. Although having a player like that dominate the scene does create chaos when he is removed. You could give the evil country a previous leader who raised it where it is now thru such diplomatic maneuvers and a current leader who tries to play the same games but fails with catastrophic consequences.
World war one was largely created because politicians were playing political games and made a web of commitments so complex nobody could control the situation. Bismarck, or anyone with any sense, would have simply reinterpreted any agreement that potentially triggers a war of that magnitude and instead negotiated a peaceful settlement that just happened to be in his advantage, but unfortunately most leaders react to complex situations by following what is expected of them so that they do not end held personally responsible for the mess. And that means following the agreements and letting the chain reaction propagate. This would be a feasible way to go MAD, I think.