Consider that you've used the switch diagram to interrupt both leads to the LED and really needed only to interrupt one. To confirm this, connect the wires as shown for 1 and 3, but connect wires 2 and 4 directly, bypassing the switch.
What should happen is that the device will energize/activate.
Swap things around and connect 2 and 4 through the switch and connect 1 and 3 together. Device will energize/activate.
What you have is a double pole single throw switch. It's convenient when you have to switch two different circuits that may have different power sources. If you have two different circuits to be switched on at the same time and have the same power source, you'd simply jumper across 3 and 4 as well as connect the power to 3 (or 4).
I suspect that you have some feature with the factory switch that is not clear in the diagram. There's no need to have a ground lead independently wired to a switch, unless there's a circuit from the 12v source (white) to ground (brown) that you will not be using.
If you are satisfied that the last paragraph applies, connect your white lead to 3 and your green lead to 1 (or use 4 and 2 respectively) to accomplish your goal.
EDIT: Allowing that you also desire to have the on-board LED to illuminate, it may be that one pair provides power to the LED. Test the circuit for both options noted above. If the LED lights up for either, it's your choice. If it lights up for only one pair, well, use that one. The crossing diodes may be that which allows for illumination, regardless of the selected pair.
OP Suggested Edit: grounding in a motor vehicle is not so much an earthing of the circuit for protection but more a means of providing the second leg of the circuit. Imagine how much of a mess you would have if you had to use a common bus throughout the vehicle or if you had to have a negative lead for every device which uses 12v power. The chassis and frame become that common bus. Any ground you can use is the same ground across the entire vehicle.