^^^If you behaved in a vile manner in your sasural and made your FIL decide that you had to leave, would he need your hubby's permission b4 kicking you out? He'd be fully within his rights to tell you to leave, either w/ or w/o your hubby (depending on whether the latter supported you). The head of a household can determine who lives there.
In this case, I'm not saying that Jyoti is wrong, as described by you. But in the Sharma household, Deepu is not the head - KK is. When Deepu cheated several months back in not taking his exams or even attending school, KK kicked him out. When Sushma eloped w/ Brij, KK banned her from coming.
Similarly, it's KK who decides who can stay, not Deepu. KK can't end Deepu's marriage, but if Poonam behaves in a way vile enough that KK decides that she doesn't belong there, he can kick her out, and add that if Deepu doesn't like it, he can go w/ her. He doesn't have to beg Deepu to do anything. Somewhat like CM threatening Sudha that she can leave w/ Jyoti, and Uday too if he chooses to back her.
Besides, at this point, what can Deepu do? He very clearly, after describing Jyoti's & Sudha's parentage and the fact that they were his mausi's daughters, warned her against ever describing them as step-sisters. And what does Poonam do? The first opportunity she gets, she says that Jyoti has no relationship w/ her. What does Deepu do in such a case? Beat her? He's done what he's supposed to.
The best thing they can do is call the Trivedis, explain to them what's happened to date, and ask them how they recommend proceeding.