I took 3 international flights this year and upon return to the US, always have to line up to clear immigration after getting an X from the APC due to my name not appearing on the manifest or doesnt match.

Background:
Green gard holder, primary residence in the US
Passport from ASEAN country
Name variation due to my country of birth not officially having a last name designation.

So in my passport, there is no distinction between first name last name, it is listed under Name as Abraham Barry Charlie Doug, and let's say Barry is my last name and Charlie Doug is my ethnic name and Abraham is my Christian name.

On my green card, my name is listed as First name: Abraham, Middle name: Charlie Doug and Last Name: Barry

How should i book my tickets for international travel?

I have tried:
Abraham Barry
Abraham Charlie Doug Barry
Abraham Barry Charlie Doug

Thanks in advance!