chapter11:
Are you the primary applicant? If so, also carry an employment letter just to be safe. I am a derivative applicant. I had also carried my previous H visas, other than the original AP document (3 copies), passport, EAD and copy of 485 Receipt Notice. The immigration officer only looked at the AP and stamped my passport with a new I-94. What's your POE? Mine was Philly and it was really cool. Hardly took 10 minutes to get through secondary inspection (that's what they call it, not a big deal).
BTW, I travelled by BA and they didn't take a copy of the AP.