I mean yeah ... if you start CO-ing "too" much, you ditch Monk.
I don't see why Monk has to be Lawful in a home game.
Lots of BBEGs in eastern fiction / fable / myth / TV (etc)
clearly qualify as a Monk-type, and yet are Not Lawful.
You bring Chaos to the empire, but you are not a Barbarian.
Warring States era. Any Dynasty collapse not involving Mongols.
Lesser Vehicle -types can be virulently anti-Deity.
... even half-Slaads do it.