Why do I think Kavya would have given her Flat documents to Anirudh and got her Divorce paper signed by him. With this she is free to marry Vanaraj and also she has no place her own to stay. Claiming fake pregnancy she might just end up in Shah house. And we know Anu will not object Kavya's entry and Baa also may not object as she might think her grand child is in Kavya's womb.
So it is either to live with Kavya or to leave the house for good for Anu and her kids.