I decided to rebuild the elevator (together with my son) and control it with BRIXO.
In order to make it work I had to make some tweaks, I used microswitches (not included in BRIXO kit) and soldered a diode on each switch. The switches are used in 'normally closed' mode.
The BRIXO battery box can easily switch polarity, so when the elevator reaches the top or bottom the switch will interrupt the current. When the polarity changes the diode will let the current pass and the elevator will move in reversed direction.
If you look at the schematics you can see the similarities in the LEGO / BRIXO section below:
Elevator in action: