Look into analytic expressions of the KKS single-phase compositions for special cases
If we do the matrix inversion analytically, we don't need to solve a linear system of equations every time step to get the single-phase compositions. This is how Trevor Keller does it in his code (in his case aided by SymPy to invert the matrix symbolically and then generate code for the resulting expressions). This gets messy for complicated situations, but could pay off for a few targeted cases.