it seems that part # 61131376191 is a male pin that would fit into the female pin from the above picture.

The male pin would have to go into the receiving part of the X609 connector (normally those male pins are integral part of the A6, trailer module). I have two conflicting diagrams for the X609, one is stating it is a 20-pin connector, and the other one indicates it is a 15-pin connector. The only sure way to find the correct information is to pull the cargo floor and take a look - i will try that this weekend, hoping that the weather is good...

An option when these two pins are pulled from the X609, the femelas cut off and the wires are further spliced to extend them to the tailgate - this is a very feasible option as well... The rear fog lights are fed into the trailer module in order to provide the rear fog ability for the trailer itself, when the two lights are fed into the A6 module, through pins 10 and 12 of the X609 and ONE feed is pulled out from the A6 module on the pin 3, traversing to the connector X393 pin 2 (the X393 is another trailer harness connector that is located on the backwall of the car, close to the grommeted hole, through which the trailer harnsess is passed in from the outside, and that harness is plugged in to the X393).

so, if the pins 10 and 12 are not coming to the trailer module, they are not being processed for the trailer fog light.
if they are not processed for the trailer fog light, no trailer fog light can be activated
chances are, if the rear fog lights are not fitted onto the car from the factory, there is no requirement for the trailer to have fog lights either.
bypassing the trailer module and wiring the pins 10 and 12 of the X609 directly to the rear fog light connectors will not have legal or operational effect on the car unless...

unless that car is being used in multiple jurisdictions where the rear fog lights are NOT required and other locations where those lights ARE required.

percentage of the vehicles in such a predicament is very low, almost non-existant...

therefore, if we don't find a connector that will mate to the X609 for a clean install, it is possible to take your proposed route and remove the pins 10 and 12 from the X609 and wire them into the rear tailgate directly. the only thing that I would propose in this case, source a complete set of one 2-pin male/female connector (could be OE BMW, or anything like 9005, 9006 connectors widely available online) and make the connection as OE as possible, without soldering and /or crimping...

Just thinking out loud... lol...