I did it!
It was quite some tedious handiwork to file the pcb by hand to avoid damaging it, but in the end it worked out, the SD-Cardreader Expansion Card is done and working:
This is how it looks installed into the Framework Laptop 16:
There might be a better solution for the port cutout, but I’m going to keep it like this for now, until I come up with a better solution. For the time being it’s: 
Mission accomplished! 