Is the application of Le Chatelier's principle wrong in this case?
Try doing the calculation using the fact that $K_x$ actually is not constant at a given temperature.  It depends on total pressure as well as temperature.  Assuming you maintain ideal gas behavior $K_p$, and only $K_p$, is constant when temperature is fixed.  In terms of that equilibrium constant you should find the reaction following Le Chatelier's Principle both of your cases.