I originally came here today to suggest an approach of powering a relay from the brake lights and wiring it across the switch. This would automatically 'press the button' the first time you applied the brakes, and you would be good for the rest of the trip. But if the jumper approach works out, it's the best! Yours for the price of an inch or two of wire.
Edit: Oops,good try, but the manual says that a subsequent press turns it back on. Back to the drawing board. Cross fingers for the super simple jumper approach in the video above.
Next Edit: How about a latching relay? But once again, cross fingers for the super simple jumper approach in the video above.