Is boycotting better, or doing what the family wants, in light of the rejection of ideas and actions that violate Sharia and their potential impact on children, especially with the existence of customs and traditions that contradict religion?

If the matter is confined to either secluding oneself from people or mingling with them at the expense of one's religion, then mingling is not permissible. However, a third path can be taken: mingling with them, winning over their hearts, influencing them, and teaching them, while patiently enduring their harm. This might lead to the triumph of the دعوت (call/mission). The matter should be made easy and efforts should be made to achieve what is in line with the شرع (Islamic law). As for Indian culture, its ruling refers back to the شرع: whatever aligns with it is permissible to adopt, and whatever contradicts it must be rejected.