Get an "HPTuners" Software and Interface-Module package, and learn how to use it.
Then You can set the default behavior any way You like,
with zero changes to the wiring or Switches.

But what you need to do first is to "beef-up" the Transmission as much as possible,
unfortunately, much of the super-mushy Transmission behavior is mechanically "built-in".
Sometimes a "Shift-Kit" will fix some of it,
which then will allow You to actually make it act like You really want it to with the Software.

I've used HPTuners on a LSx swap I did and it has a HUGE learning curve and you still can't tune a vehicle properly without a dyno. This is a 2017 BMW M240i and HPTuners does not support BMW's, there is no way to make the car default to sport mode. The stock transmission in these cars is capable of being behind an engine with over 700hp so it doesn't need to be modified. I'm merely wanting to have the car default to sport mode, I'm not looking to add a bunch of HP.

 

The Stock Transmission would completely melt-down behind a 700hp Engine without serious modifications.
HPTuners may well not cover BMW, but there are certainly other Tuning Softwares that will.
"Sport-Mode" does not "increase-power", it just changes the way that the Transmission shifts,
and the aggressiveness of the Electronic-Throttle-Controller,
and possibly the Electronic-Shocks (if so equipped).

( I used to be the "Problem-Child" trouble-shooter at a huge 14-bay BMW Dealership,
( South-Motors-BMW, Miami ),
and found that the biggest problem with BMWs usually turns-out to be their uber-fussy owners )
.
.
.

 

Look up the ZF 8HP transmission and I think you'll be pleasantly surprised. I'm well aware that sport mode doesn't increase power, it does make a huge difference in throttle response and shift points. It can make the suspension stiffer too, but I have mine set to where it has the same ride quality as in comfort mode. There's a few different tuning apps that BMW owner's use, but none of them allow for the car to default to sport mode. Again I'm just wanting the car to default to sport mode, not to spend $400-$700 on tuning software to get more power.

 

as an answer to your original question - you should find out / consider the following

• system response - when the button is held down continuosly or for the longer period
• system response - when the car is started and the motor dies off (as "taking off" on the slope)
• the type of the button used : resistive , other
• the voltage the button switches
• the electrical isolation of the button circuit from the rest of the car's electrical system
• the possibility of bypassing the button without altering the ® circuitry (so you can disconnect it and the ® circuit'll function as prev.-ly)

the solution depends on some of the information revealed by the above list

 

The used switch I bought came in. I haven’t had time to hook my meter up to it or the harness in the car, but here’s some images of it. When pressed, the momentary rocker switch has two tabs on each side that make contact with the board. The first image that has four square contacts on the board is where the switch makes contact. I’m wanting to send a pulse to the upper two contacts at the same time, but would like to know if this is possible to do at the pins instead of someone having to take apart the entire switch.

 

the board seems to have resistors on it - from were 1 could speculate that the decoder has either resistive or voltage range input

the rubber contacts may have resistance from several ohms to couple of kΩ

the 1-s idea is to use the pass gates with equivalent series resistance for the conductive rubber . . . but since the passgates have DG SG leakage capacitance - it's not 100% sure the proposed schema would reliably work