regarding intercaste marriage in Kerala

Yes, in an intercaste marriage, child belongs to the father's community it was brought up in the father's household. As regards to inheriting family name, it has been old practice; this is no longer i vogue.

I belong to Kerala.

Your case is peculiar.  As per SC decisions in this matter, the child will carry father's caste.  But SC in some cases granted the caste of mother to the child on the ground because the child was born and brought up in the atmosphere and culture of mother.

Now, why your case is peculiar because, as you are saying that in Nair community, matriarchal system prevails.  As such, the child can carry the mother's caste.  If you are interested to bestow the child the mother's caste, then certainly you can proceed in court of law and get a declaration that your child belongs to SC. If you want your child should carry Nair, then it is upto you.  You need not do anything.

Matriarchal system no longer prevails among Nairs, or any other caste or community in Kerala. The child can claim to belong to the mother only if brought up in the community/household of the mother.

If you are interested to know the practical example study in google on Rohit Vemula case, where parents are both BCs, but he claimed that he was brought up in SC community and was issued such certificate.