Overall, with a valid AP you should not have any issue with CBP. The biggest issues I have heard of is with airlines not recognizing AP as a valid form for entering the U.S. here are a couple of links that may be helpful. The CBP Carrier Guide clearly lists the AP Card (form I-512) as acceptable to board someone to the U.S., so it is a good idea to have a copy of the guide with you when traveling.
Good Luck!
https://www.cbp.gov/sites/default/files/assets/documents/2023-Nov/Carrier Information Guide ENGLISH.pdf