Not if you modelled them accurately: very few templars, hospitalers etc. knights were readily available at all times (for instance, the disaster at Hattin has costed the Templars 1/3 of the total brothers knights - around 230 men - meaning they had less than 700 knights, in addition of course to the sergeants and squires. No to mention that the Templars were usually the most numerous of the orders at that period). So, if you have a unit that costs 2X the amount of the best knight unit of that faction, a high maintenance cost and smaller unit size, and takes 2X the turns of a regular knightly unit... no, they won't be invinsible.
The knight orders were indeed kind of independant (the Papal control was only a formality, they usually manipulated the Popes to extract favorable Bulls, not vice versa) but they held only castles and outposts, they had no state in place.