Our friends did this a few days before we flew home, and we did the same— it worked for both of us: arrive at STT before the airline desk opens. There aren’t that many flights, and many airlines don’t open their desks very early. For example United on Sundays doesn’t open until 11 am! We were the first people in line at United and then sailed through customs. We had to wait two hours past security for our flight, but it was fine.