You're right: this is a peculiar animal -- quasi-military, quasi-religious; not wholly either one, and layered on that is their status as a sovereign entity among the sovereign nations of the world.  The Wikipedia article does a decent job in describing the history and the organization of the order. 
I think you have pinpointed the critical issue: Did the Patron of the order misrepresent the pope, or fail to represent him, at a critical meeting of the hierarchy of the order? The pope can and probably will remove the patron of the order; he can also transfer the prelate of the order, and withdraw or transfer the clergy who are attached to the order.