usanep, not a legal advice, from what I have read, you can create a travel authorization for your child to be with relatives. Its a bit of an hassle, but you could have someone come to India and fly along with you or in another flight same day or ever earlier than you guys. We have a open border but authorities may ask to see your citizenship.
Here's a sample:
https://www.us-passport-service-guide.com/minor-travel-consent-form.html#minorconsentform