There is absolutely nothing wrong in having Twinkle keep her last name. If she feels she gains her identity from keeping her surname as Taneja, then that is absolutely acceptable. Kunj shouldn't force her to change it to Sarna when she will not do anything to uphold it. Sarna is a name that Kunj and his family are proud of and will do everything to honor it. Twinkle, on the other hand, has just arrived into the family and although she will do her best to be a good daughter-in-law, her primary loyalty will always be to the Tanejas. That is who she has lived by for the past twenty plus years and that is a feat that will be really hard to erase.
On the other hand, she has married into a new household. Her every action is going to affect the family whether she wants it to or not. Not to mention, Taneja and Sarna are two highly prolific names. The media waits for the chance to jump at these type of occasions so that they could increase their own reputation by sullying others. I understand Twinkle's inhibitions of being associated with Kunj. But the fact is that she married him out of her choice. No one put a gun to her head and made her do it. She chose to go through with it despite there being so many opportunities for her to pull out. So now she has a moral responsibility of keeping the image of being Mrs. Sarna. By refusing to acknowledge Kunj as her husband to a legal authority, she is denouncing the Sarna name and her affiliations to it. She has not entered the home mentally and that is why when she makes the propositions of her being determined to be a good daughter-in-law, it comes out being really fake.
She is not Mrs. Sarna right now and in order for her to become one, it is going to take a long, long time. She is not ready to accept that title emotionally, mentally or physically. She is not ready to accept this new identity and she is not ready to fulfill any of the responsibilities that come along with it despite her saying so otherwise. If this is the case, then I truly don't understand as to why she got married to Kunj in the first place. She has truly ruined his life as he will never be able to get the respect, support and love that he deserves as a husband. Usha will constantly be interfering in their marital life because she knows that what Twinkle is not the right woman for him. At least not now.
By getting married to Kunj, Twinkle made a truly terrible mistake.