Your query has three important sub-questions. These need legal clarifications and not emotional views.
A.Non-claiming of reservation tag.
The reservation quota or BC tag is not mandatory to be taken. It is allocated provided the individual applies for it with adequate proof. This issue will crop up only if the adults or children enter the caste column as BC or SC or ST.
So, please check with the school admission forms, have you or your husband entered the caste section or just mentioned GM.
You can bring up the child in whatever religion or caste-based traditions or rituals who believe or prefer.
B.Carrying surname of Brahmin caste
Generally, names have first name and surname OR first, middle, last name. Often surname would be that of the father's name given by parents be it, boy or girl. So, your children would not have Master X Swati or Miss Y Swati. I'm sure you would have entered your husband's name as their surnames. So, if you want the children to have your surname, then it would involve a formal name change. Please discuss this with a good civil lawyer who is good in intercaste inheritances and rules.
C.Inheriting mother's caste.
In general, the children carry the caste which is that of the father. There are many court rulings that are individualized.
Marriage does not change the caste of the adults involved but can change the caste of the children based on how they are brought up.
In Inter-caste marriages, the legal view would be that children should be given the right to decide on the caste after they turn 18 until then parents decide.
I'm not allowed to give links directly, please check these three reports that relate to upper caste and lower caste unions and it's relevance on children.
So, I would suggest, please have a serious discussion and then contact a good lawyer who can help you out. Also, please check on the impact of your decision for the Proof of Identity documents that you and children have. In my view, point A in my answer will not matter much but point B and C will need a good legal eye to guide you appropriately. Also please check the impact of your decision, if some blood relative of your husband files a case for a share in the properties that you both currently have.