Mogegg, the TSM (turn signal and headlamp module) does control headlamp functions as mentioned.
The fusing is sometimes after the switch that selects the function the fuse protects. As already explained the dip headlamp fuse will not be live until power is received from relay, and this won't happen whilst relay coil is energised leaving headlamp main beam selected. To prove this with ignition and headlamp main beam on you will find fuse 16 live but fuse 17 dead.
Don't have my M3W anymore to check but dip/main relay is the one with all five pins connected and orange/blue control cable.
I think the redundant cable(s) in headlamp shell may be for position lights which are not used in US models I believe.
It appears you don't have M3W parts book which contains engine and chassis wiring schematics. Suggest downloading
here.
