I'm having serious problems encountering Ameli in the hallways after she joins and leaves the other 3 clubs, is there any way to influence the game to increase the probability of it triggering, or am I stuck waiting to have luck? How could I modify the event to have an easier time getting it activated?

Looking at the event in Location\School\Hallway Shared\MaidClubStart...

Triggers to start:
1) Player has to be in a "Hallway Shared" room, which are: Upper/Ground floor West/East, plus Annex Wing and School Entrance.
2) Must be during a school-time break (if you see her in the hall during class it doesn't count)
3) Ameli hereself must be in a room with "Annex" or "Floor" in the name. So School Entrance doesn't count, but "Coppers Floor" does (can students even go home during break?).

No stat checks at all, but you have to trigger it four times in total and the event forces a 14-21 day wait before the event will trigger again. So the absolute soonest you can get Maid Club started is six weeks.

(edit): You can edit the event itself in the visual events editor, but note if you do that during a save you have to go into Debug -> Reload Events for the change to take effect, and even then that doesn't change when the event is currently scheduled, just makes it so repeats after that pop faster.