Referring to the original question: does the aircraft eventually regain the localiser, once it has departed, or does it remain off course?
HC, is your FMS of FW origin, and if so, is there no opportunity to subscribe to a larger database? We use the Universal UNS-1E, in the S-92, with a worldwide airport database, including all SIDS, STARS and Approaches (including the transitions and Missed Approaches). It even allows us to fly the fully Managed LNAV/VNAV NPA approaches TOD mentions, allowing CDFA stabilised procedures, rather than 'dive and drive'.
TOD, our FMS sets the course too, but only for NPAs, not for ILS.
The S-92 coupled ILS function seems carefully programmed to closely resemble a badly hand flown ILS, with even a 6 kt crosswind capable of generating in excess of half scale localiser deflection close to minima (flashing yellow diamond: just what you want in IMC)!
Some changes required before looking at Cat II I would say!