I suggest that you buy the ticket for infant as a child ticket and DoB is not mandatory. There will be no issue with Indigo at the time of check-in. I am 100% sure, no one will even ask for baby age at check-in.

Hand baggage with baby stuff / Diaper bag is allowed for infants as carry on. I am sure about it.

Overall, I have traveled with an infant along with my 4 yr old daughter both domestic and international and never felt the need to buy a separate seat for infant. But you may have your reasons...