if Karin helped Sakura give birth she could have just taken it without them noticing.
This does open up the possibility of Karin being the mom for sure but its far from confirmed.
Let's be honest here Karin is pretty weird when it comes to Sasuke. Would it be that hard to imagine her holding on to the cord?Suigetsu thinks it is hers but he is just assuming that. There is no concrete evidence that it is hers other than the fact it was in her desk.
It just seems like to much of a set-up for the inevitable "got ya". Suigestsu not knowing for sure where it came from, the fishy convo with naruto ect.
I am open to the idea that Karin is the mom now, but i still am fairly certain this is a red herring.