Obtaining caste certificate

You can apply to the authorities informing them of the factual aspects. It is for the authorities consider and decide.

The father's caste will be the children's caste, but not the mother's.  

Biological  fathers  caste  will be  Children  caste  even if  mother  got  divorce 

No doubt traditionally a child's caste is determined by the father's caste, in inter-caste marriages, a child can choose to be recognized under the caste of either parent.

his choice is often formalized through a community certificate.

in the case of inter-caste marriages, where parents belong to different castes, the situation becomes more complex. 

The choice is often based on a declaration made by the parent or parents, stating which caste the child will be identified with. 

While Tamil Nadu has this specific provision, other regions in India might have different practices or interpretations regarding the caste of children in inter-caste marriages. 

Apply to get said certificate in concerned authority office. 

It is not only  a general tradional but a matter oflaw that child is entitled to the cast status of his father only.