Your second wife is not biological mother of the son born out of your first marriage 

further she is not adoptive mother of your son 

sje would not be eligible to avail free education for her step son 

You want to submit proof of custody of  child to obtain admission in school. Notarized affidavit is not accepted as proof of biological motherhood. But you can produce proof dissolved marriage with ex- wife, proof of  divorce  and proof of  second marriage to establish he is your son and teacher is  your wife. Other option you have is to go in for step child adoption with the consent of biological mother and willingness of second wife. For adoption you need to first register with Central Adoption Resource Authority, Government of India, Ministry of Woman and Child Welfare. After adoption seek approval of Chief Judge, City Civil Court, Hyderabad. Court will direct GHMC  a birth certificate showing present wife as mother and you as father. On the basis of birth certificate you can obtain all privileges available to your wife including admission. After registrar with CARA it takes two months to obtain approval of Court.

yes notarized affidavit will be enough in above case

The custody of child will not entitle your 2nd wife to have an y rights over the step child.

If she wants to avail the privileges that is available as a school teacher towards the education of her own children, then she has to get the child adopted and she should become the adoptive mother of your child born out of your first marriage.

Whether you get the custody of your child or not, you are the natural guardian of your biological son, hence the transfer of temporary  guardianship of your child to you will not make any difference.

You can get your child under your custody and you may have to spend for your child and bear all his expenses as any other father of the child, your second wife may not be able to get any privileges towards the step child.

Dear Client,

As per the facts which have been provided, you can send a letter to the school to ask if such kind of provision is there and whether they will accept the affidavit given by your first wife.

Hope this clarifies your query and requirement.

Thank you.

Dear Client,

You may consult an advocate to go ahead with the adoption

Thank You.

Dear Client,

Under CARA rules, an adoption within India should cost no more than Rs 46,000: registration for Rs 1,000, the home study process for Rs 5,000 and Rs 40,000 for the agency's official child-care corpus fund. (Adoptions by non-Indian parents have a separate, higher fee structure.)

Thank You.

If you adopt him and take benefits of second wife then he will have to waive all benefits of first wife his biological mother

You can contact the chosen advocate directly by getting his contact details which are available in this forum and clarify all your doubts from him 

Consent of biological mother is must for deed of adoption to be executed 

you can contact any  lawyer on this website in this regard 

legal fees vary depending upon lawyer engaged by you 

As your Second Wife is neither the Biological Mother nor the Guardian of the Child, you won’t be able to get this benefit provided by the school.
If you are really serious about pursuing this, I would suggest you and your wife either adopt the child officially (if this needs to be permanent) or you and your second wife jointly file an application in court for guardianship of the child, which would obviously not be challenged by your first wife because she’s on the same page as you.

Hope this helps.