You should. That's the way it's done. The original BC is sealed, and the kid gets a new one with the adopted parent's name. It's why Bill Clinton is Bill Clinton, and not Billy Blythe. It's why Gerald Ford is Gerald Ford and not Leslie King. Clinton and Ford are their LEGAL names. The adoptor is their legal parent.
When the guy was adopted, his legal name became Barry Soetoro, or something like that (Soebarkah, etc). That means to change it back, he would have had to go to court, which he clearly did not. He has, therefore, in the opinion of some very intelligent people, and me,been cruising under a name not legally his.
To top it off, he lied about on his Illinois Bar App. Perhaps not coincidentally, he is no longer allowed to practice law. This name thing is very serious business, indeed.
If it is proven, the question becomes, "what among his various actions is invalidated?"
I remember threads from back when Clinton was President on whether Bill Blythe was Clinton's real biological father or not. Like Barack Obama Sr., Bill Blythe had another wife from whom he was not divorced when he married the future President's mother.