a monk can only be lawgul, you would have to ditch that part as a ninja variant of the monk.so were ninja's. Instead of temples they had clans. Ninja traditionaly had chain of commands, rules of dealing with employers, they are just as lawful as monk.
which is why the ninja as a rogue alternative, is damn wierd. han solo is never becoming a damn ninja , ever
In Japan mythos/lore and in respect to their application in Pathfinder I'd say ninjas are chaotic evil. The Oniwaban are a good place to start drawing the lines. They are far from Shaolin et al.