I don’t think so. After all the mom is the same, presumably. It would be an adoption by Lolo, not a claim of fatherhood. Besides, Fukino says she has Obama’s original birth certificate on file “in accordance with state policies and prcedures”...whatever that means.
This was what the poster was talking about here on FR..he was adopted by his step-father and his original birth certificate was sealed.
In that context, the latest version is considered "the original", since any others are sealed, and can only be revealed, even the person concerned, by court order.