Update: I made it back to the US on my original flight. Since I booked my flight using points, I didn't have anything to lose and decided to push my luck. I checked in online, put my A-number in the form and entered May 2027 as the expiration date of my greencard. Also added my Global Entry number. Went to the airport and nobody asked for the physical greencard at the bag drop off. There was one more document check but nobody wanted to see the actual greencard or the letter so I got on the flight without any issues. At immigration in the US I got in the Global Entry line as always. The officer asked for my greencard and I said I don't have it with me but showed him the extension letter. He looked at it and said 'Welcome to the US!"... I didn't even have to go to secondary inspection so I must have gotten incredibly lucky.
I had a plan B in place in case they would have denied boarding which I totally expected: I would have taken a flight to Mexico City a few hours after my original departure and met my husband there. He was ready to travel there from the US just to deliver my greencard and to get back together to the US on the next flight.
Thanks again for all your replies. I am incredibly grateful for what happened today and that was absolutely not what I expected to happen. Success factors for me were online check-in and Global Entry I believe. Plus a lot of luck! I don't ever want to go through this again though.
