Upvote:1
I haven't been following this very closely, but several months ago it was apparent that travelers trying to use ESTA to fly from Europe to the US were having their ESTAs cancelled automatically even if they were excepted from the public health restrictions.
You will therefore need to find out how to prove your relationship before you board, whether by getting your exempt status formally registered in the ESTA system or by getting a B-2 visa. Once you've taken one of these steps with the US government, you can bring whatever documents you needed for that along with you to show to American Airlines.
If you already have a valid visa that was issued before the travel restrictions were imposed, you might want to ask the consulate what documents they would require to issue a visa now.