This Maharashtrian ritual of changing name of the bride actually takes place after the Saptapadi or phera where the groom chooses a name for his wife by writing it on a plate of rice. In today's times, many people skip the ceremony in the sense that they do choose a name but it's not officially changed on documents if the lady wishes to retain her original name (not surname).
Bhavani insisting on this ceremony after not wanting to do anything is her way of getting back at Sai, knowing how fiercely proud she is of her roots. Guess that Bhavani knows Sai will refuse to change the name (I hope she also retains her maiden surname but seems unlikely). So the Kaku and her Makad Gang is going to keep provoking Sai's pride and extracting reactions out of her. Virat will obviously write Sai on the rice plate and say he has picked that name for her.
How many of us want her to continue as Sai Kamal Joshi?