It could be that Claire decided in front of witnesses and even wrote up something in her journal that if anything were to happen to her on the island and she was unable to care for Aaron that she wanted Kate to take care of him.. That could be why once they were off the island she was able to keep him.. It was at his mothers wishes...

__________________
Please don't give up, Des. Because all we really need to survive is one person who truly loves us. And you have her. I will wait for
you. Always. I love you, Pen

DESMOND: I love you, Penny. I've always loved you. I'm so sorry. I love you!
PENNY: I love you too.
DESMOND: I don't know where I am, but--
PENNY: I'll find you, Des--
DESMOND: --I promise--
PENNY: --no matter what--
DESMOND: --I'll come back to you--
PENNY: --I won't give up--
BOTH: I promise. I love you.