Consider this - She and Lolo were getting married and moving to Indonesia. Lolo had to adopt the kid to get him into school, etc. He couldn't do that unless there was some sort of legal paperwork to show the Indonesian officials. Remember, the divorce and 2nd marriage were done within days. Actually, there was some discrepancy in SADO's passport applications of the marriage date vs divorce so that was another oopsie in trying to keep the lies straight. So, Ann gets a "divorce" to make Indonesia and the US authorities happy. Then a few years later when she ships the 10 year old back to HI, the Dunhams have to begrudgingly host Sr.'s Magical Mystery Return to repair the bogus paper trail.
There’s much in what you say that makes sense, and one thing in particular stands out; she couldn’t take him anywhere or be party to an adoption without custody.
AND THE DIVORCE GAVE HER THAT.
It also made the ‘birthdate’ official. In the absence of a birth certificate - that was all she had.