Salaam alaikum, and thank you for your question!
I want to start with the disclaimer that I am neither a scholar nor qualified to give fatwas on whether something is halal or haram.
For specific cases like the one you have mentioned, it is best to ask a scholar whom you trust and who is also familiar with Indian culture so that they can give a more nuanced, culturally sensitive answer.
However, insha’Allah I can give a basic answer to your question.
Why Use Father’s Surname?
In general, Islam teaches that children should keep the surnames of their fathers rather than their mothers.
This is based on the passage in the Qur’an where Allah is talking about adopted children:
Call them after their fathers. That is closer to justice in Allah’s sight. And if you do not know who their fathers were then they are your siblings in the religion and people under your patronage. You are not to blame for any honest mistake you make but only for what your hearts premeditate. Allah is Ever-Forgiving, Most Merciful. (Qur’an 33:5)
Inheritance and Marriage
The primary reason for keeping the father’s surname is to preserve knowledge of lineage.
Knowing who one’s blood relatives are is essential in both issues of inheritance (since people receive a certain inheritance based on their relationship to the deceased person) and marriage (so that one doesn’t unknowingly marry a mahram).
Having everyone use their father’s surname keeps things uniform, rather than having some people with their father’s surname and some people with their mother’s surname.
It also makes it more difficult for fathers to ever deny their relationship to their children (which isn’t as much of an issue with mothers, since anyone that saw her pregnant would know that the child belongs to her).
A Couple Exceptions
The only case where this is not required is if the person’s father is unknown, and then the person can choose their own surname to use.
Some scholars also say that if a child is born out of wedlock (i.e. their mother and father were not married when the child was conceived), then they are to be attributed to the mother rather than the father.
Other scholars say that even in this case the child is to keep their father’s surname, so long as the identity of the father is sure.
In the case you described, the father is known to the child and was married to the child’s mother at the time of conception, so therefore the child should use the father’s surname rather than the mother’s surname.
Rejecting Class Divisions
I commend you for rejecting the caste system that is not only prevalent in India, but throughout the world.
It may go by different names, but the division of people based on their socioeconomic class and/or lineage is universal, and universally harmful.
Allah has told us that these differences ultimately don’t matter, and instead we should focus on improving our character:
O mankind, indeed We have created you from male and female and made you peoples and tribes that you may know one another. Indeed, the most noble of you in the sight of Allah is the most righteous of you. Indeed, Allah is Knowing and Acquainted. (Qur’an 49:13)
A descendent of the Prophet (peace and blessings be upon him) may have “noble lineage”, but that can never be a substitute for believing and striving to be the best Muslim one can be.
Islam is meant for everyone, and everyone has the chance to reach the highest ranks of nearness to Allah, regardless of their lineage or past history.
A Strong Statement
I think it would be a very strong statement of rejecting the caste system to go by one’s father’s name even if it isn’t as “noble” as one’s mother’s surname.
Know that those who make fun of one’s surname for being “less than” are demonstrating ignorance.
If someone is being given a hard time because of it, they may want to gently tell the others that surnames are not an indication of one’s worth or character, and to please stop their teasing or criticism.
Insha’Allah, these actions may encourage others to reconsider their own views of the caste system/class divisions.