I believe this will depend on how much the index change is. If it can change the mode effective index on the 4 or 5th digits, you can use FDE to analyse the mode property. If the index change is small and need long section to reflect such change, 3D FDTD will be required.
Foe one specific mode, you can use symmetry BCs to reduce the simulation volume and thus increase the simulation speed: Symmetric and anti-symmetric BCs in FDTD and MODE
If it is for singlw wavelength or small wavelength range, you can use EME.
exiting more than one mode will depend on the solver you use:
EME can get S parameters with several different modes, as long as you specify the modes in the ports.
If you use FDTD and port, only one mode at a time. You may need to consider the physical meaning of multi-mode injection by use of several mode sources.