I believe it's a motion sensor for the alarm system.

The "inverted dome thingie" is the housing for the Alarm System Control
Module.

Your car has Central Locking, and there is an Alarm Kit that you can buy and
install yourself. The horn to the alram gets installed in the area to the
right (passenger side -- USA cars) of the microfilter, the Tilt Sensor goes
in the trunk, near the fuel filler, and the control module that goes into
the dome on the headliner.

After you install the alarm components, you have to visit the dealership, or
an independent BMW garage, to have the system programmed so the car will see
it.