I did twice with AP

I was I-140 approved and I-485 pending while I used AP. You just need AP and passport. But I kept the other document copies with me, such as I-140 approval letter and I-485 pending notice, etc. You can take as much as you can. they did not check these when I entered. You have to wait in another line for people with AP. It takes a while and no problem for me. I also took my kids with me. They were born here with American passports. Good luck.