I don't think there's an issue for applying a passport for your kid. You need the following documents for applying kids passport.

1. Application duly filled and signed by father & mother
2. Passport copies of father & mother
3. Child's berth certificate in original
4. A demand draft for Rs.600 (minor) in the name of
concerned passport office.
5. 2 passport size photo of the child

You have to carry all the original documents with you at the time of application. They will ask the same for verification. Also carry whatever documents that you have to prove you & your wife are a native of Kerala.

I got my son's passport in 7 days. My wife was having a Delhi issued Passport and me was having a Kerala issued Passport. No questions were asked. The only difficulty i faced was; i never knew that i had to take my kid along with. So i was asked to come together with my wife and kid.

So take your wife and kid/s along with... Gud Luck.