Originally Posted by Alex71

Thank you squeakr and thank you, everyone, who provided helpful tips to resolve the situation.

Here's the final update on how things played out: we arrived at LAS three hours before the flight at the time that check-in was scheduled to open so that we would have plenty of time to resolve potential problems.

The check-in agent looked at our documents and the letters from the consulate, took everything with her and disappeared for about 15 minutes before she returned and told us that everything was ok. She could apparently see that the kids had traveled to the U.S. on American passports since she told us they did not have to fill out a departure form for U.S. citizens as they left the country on German travel documents. No issue was made of it, though.

At security, the TSA agent gave me a bit of a hard time and initially told me I needed a passport and could not travel on my German identity card. He was not concerned about the kids, though. "The kids don't need ID to travel, it's his stuff that looks weird!" He called in a second colleague to debate my case, played around with my ID card in a device with UV light, read the letters from the consulate, asked me a few questions, such as my birth date, and eventually let me through.