You won't find it in a list of synonyms for "break", because lists of synonyms focus on just one sense of a word. The word "break" has many different senses, and "breaking bad" combines several unusual ones at once.
Basically, in the series title, it means "becoming bad": becoming a bad person or engaging in bad or violent behavior. But "break" in this sense has much richer connotations than "become"—which, of course, is why the show's creator chose to call it Breaking Bad and not Becoming Bad. Explaining connotations is hard, but here goes.
As your list of synonyms suggests, the primary sense of "break" is: "to sever into distinct parts by sudden application of force; to part by violence" [OED]. An extended sense is to break through a barrier, especially at the start or the end of a long process. To break ground is to begin digging into the ground, as the first step of constructing a building, or to begin digging in preparation to plant seeds in farmland, or metaphorically to undertake the first step of anything. When a pregnant woman breaks water, this means that the sac that holds the baby and amniotic fluid has burst, signifying that she will soon begin labor. Near a shoreline, a surf is said to break when it nears the shore and topples over.
Those extended senses all suggest some kind of important transition, where something "breaks" in the primary sense. Here are some more, which are closer to "breaking bad". To break free is to cross a barrier or burst through a bond that had previously constrained you; to escape. To break news is to tell someone the news for the first time. When the news itself is breaking, it's happening right now. When a storm breaks, a downpour of rain suddenly begins. The break of day is the moment when the Sun first appears above the horizon. To break even is to have your gains and your losses come out equal ("even"). If you are playing billiards, and the ball breaks left, that means it turns to the left instead of going straight.
Now are you starting to see how "break" stretches from its primary meaning toward something like "become"? The senses along the way are not completely forgotten; they're still suggested or evoked. This Time magazine article quotes Vince Gilligan, the lead writer of Breaking Bad, defining it as a Southern U.S. expression meaning "raising hell". But as the article notes, Gilligan's definition doesn't do justice to the full meaning of the term, which is probably impossible to define precisely. In the first episode, Jesse uses the phrase to refer to Walt's inexplicable change from a straitlaced, middle-aged high-school teacher into a budding criminal. That's the theme of the entire series, of course.
The OED offers this as definition 39c:
To happen, occur. slang (orig. U.S.)
and includes these illustrations:
"Things are breaking right for me."
"Everything'll break good."
So, breaking bad suggests something of the change of course of "breaking left" in pool, something of the transition into chaos of a "breaking" surf, and something of the end result as in "breaking even".
I can see why a translator would choose not to translate it, favoring instead a completely different title. What makes the title so expressive in English—the simultaneous resonance of many senses of break—makes it impossible to translate directly to another language.
