@PratikD Looks like exit permit comes with a 2 weeks window for you to leave India
My friend visited FRRO and talked to immigration officer at Mumbai airport. Both said the same thing - Come to India on Indian passport, surrender passport at RPO, get exit permit from FRRO.
FRRO said exit permit usually takes 1 week to process, but if there is a delay in hearing back from Police or Election Commission,it could take more time.
About stay visa - FRRO officer said most likely it will not be required, but since it is a new situation for them as well, they are not sure. I have looked at the FRRO exit permits online. All of them have a field called "visa" which shows the visa on which you entered India. I am not sure what FRRO will put in that field for us.
I have decided to take the leap and travel to India. Hopefully, things go smoothly at every step.